Directed Cooperation

University of New South Wales

Author: Supervisor: Yiyuan Xie Benoit Julien

A thesis submitted in fulﬁlment of the requirements for the degree of Doctor of Philosophy

in the

Australian School of Business School of Economics

April 2015

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Date ……………………………………………...... “The ball I threw while playing in the park, has not yet reached the ground.”

Dylan Thomas

Acknowledgements

This note of acknowledgment must begin with an expression of my immense debt to my supervisor, A. Professor Benoit Julien, who not only taught and inspired me into the ﬁeld of directed search theory, but also gave me invaluable advice and encouragement. He spent hours every week instructing me through theory papers, those from which my knowledge of economics has come from.

I am immensely grateful to my co-supervisor, Dr. Alberto Motta for invaluable comments and suggestions. He has provided a considerable amount of comments about each of my presentations, which I have found to be very beneﬁcial to my future career.

Quite a few scholars have provided me with important instructions and suggestions through various conferences. They include Professor Kenneth Burdett, Ran- dall Wright, Nejat Anbarci, Hodaka Morita, Kunal Sengupta, Richard Holden, Bill Schworm, Arghya Ghosh and Pedro Gomis-Porqueras, Dr. Carlos Pimienta, Ben Greiner, Jin Yu, Liang Wang and Jonathan Newton. I thank all of them.

Many debts have accumulated in these years. I thank all my fellow colleagues for their beneﬁcial comments: Allen Lau, Chunxiao Xu, Chunzhou Mu, Chengsi Wang, Jianfei Shen, Gaoyun Yan and Le Zhang; I am truly and deeply indebted to my best friend, Chirag Mohanty, who provided invaluable support.

My research is jointly sponsored by Chinese Scholarship Council and Australian School of Business, I gratefully thank them for making this happen.

I acknowledge the debt to my parents for their support, understanding and enormous sacriﬁce. I owe them too much.

No words suﬃce to thank my ﬁancee, Wen Du. All I can say is that without her everything would not have been worthwhile.

Yiyuan Xie, Sydney

Contents

Acknowledgements iv

Contents vi

Symbols viii

1 Introduction1

2 A Cooperative Game with Coordination Friction8 2.1 Introduction...... 9 2.2 The Model...... 11 2.3 The Optimum Contract...... 15 2.4 The Equilibrium...... 18 2.5 Examples...... 20 2.6 Conclusion...... 25

3 Pricing and Quality Provision of a Frictional Market 26 3.1 Introduction...... 27 3.1.1 Related Research...... 28 3.2 The Model...... 29 3.2.1 The 2 × 2 Case...... 30 3.2.2 The N × M Case...... 32 3.2.3 Large Market...... 34 3.3 Characterization...... 36 3.3.1 Comparative Statics...... 37 3.3.2 Matching...... 37 3.3.3 Free Entry...... 37 3.4 Application...... 39 3.5 Concluding Remarks...... 40

4 Competitive Cooperation 42 4.1 Introduction...... 43 4.2 Preliminaries...... 46

vi Contents vii

4.2.1 On Commitment...... 48 4.2.2 Local Competition: Bidding Auction...... 49 4.3 The Model...... 50 4.3.1 The 2 × 2 Case...... 50 4.3.2 The N × M Case...... 53 4.3.3 Large Market...... 56 4.4 Characterizing Equilibrium...... 57 4.4.1 Comparative Statics...... 58 4.4.2 Matching...... 59 4.4.3 Under-investment: Hold-up Problems in Directed Search 61 4.4.4 Free Entry...... 63 4.5 Application...... 64 4.6 Concluding Remarks...... 67

A Mathematical Proofs 70

Bibliography 84 Symbols

S Set of sellers B Set of buyers M Number of sellers N Number of buyers V Seller’s payoﬀ U Buyer’s payoﬀ X Joint action set

x Seller’s eﬀort level y Buyer’s eﬀort level p Price level

α Bargaining power Ω Buyer’s matching probability λ Seller’s matching probability

viii

For my parents, and to Wen

Chapter 1

Introduction

Since its inception, search theory has been proven to be extremely successful in explaining labor market transitions, and distribution of wages across in- dividuals and over time. In general, the basic rationality in random search literature is that agents’ choices to enter or quit the market rely on the trade- oﬀ between the beneﬁt from matching and the leisure of being unemployed, where the possibility of matching, the natural rate of separation, along with the equilibrium distribution of surplus formulate the equilibrium.

Naturally, one of the central problems in search literature is the wage determination, since it plays a crucial role in equilibrium formulating. In early random search models, this is given by an exogenously assigned Nash bargaining solution. For example, Diamond [8] used a simple search technology and the Nash bargaining solution and derives the steady state equilibrium negotiated wage as a function of the equilibrium unemployment and vacancy rates.1.In Mortensen and Pissarides [22], the Nash bargaining solution has been used to endogenize job creation and destruction. In monetary search models such as Lagos and Wright [15], Pissarides [34], Trejos and Wright [42]2, buyers and

1For a comprehensive survey of random search models of the labor market, see Rogerson et al. [35] 2Nosal and Rocheteau [25]Provided a comprehensive survey about this group of monetary search papers

1 Introduction 2 sellers who randomly meet are assumed to conduct negotiations in a bargaining fashion, so that it also derives the Nash bargaining solution in equilibrium; though the respective bargaining powers are still exogenous.

Nash [23] did not actually analyze the bargaining process, rather, he took as given four simple axioms and showed that his solution is the unique outcome that satisﬁes the four. A few years later, Nash [24] provided a simple program to verify his solution. However, his own ”smoothing” technology has been criticized by many scholars and not often deemed to be reasonable. 3, Harsanyi and Selten [11] proposed and axiomatically characterized the asymmetric Nash bargaining solutions, in which they verify that the unique bargaining solution that is strictly individual rational is one of the asymmetric Nash solutions, though they varies from each other by relative bargaining powers. The explicit bargaining process as an extensive game has been formally established by Rubinstein [37], where time preference plays a key role in equilibrium allocation. Setting bargaining power as exogenous obviously simpliﬁes the problem and is a practical choice in many ways. However, its shortcomings are obvious as well. First, when bargaining power is set as exogenous, it becomes a parametric issue to evaluate the value and much relies on empirical calibration. Second, it ignores the interaction between matching and bargaining power which gives rise to endogenous problems and weakens the robustness of the model.

Partially to overcome these shortcomings, and to address the curiosity that people have about the interactions of these models from a micro prospect; models of directed search have been established to capture the frictions in the market with a micro-founded approach, where price announcement and the choice of trading mechanism play a signiﬁcant role in allocation.

Though directed search models share the same methodology to model decentralized markets, they generally separate into the following two strands in

3Luce and Raiffa [17] described it as ”a completely artificial mathematical escape from the troublesome non-uniqueness”, while Schelling [39] argued that the smoothing technology is ”in no sense logically necessary” Introduction 3 terms of motivation: the mechanism design strand which focuses on how a stable equilibrium can be derived under a certain mechanism, and how efficient the properties of such equilibrium are; and the macro implement strand which is mostly concerned with how the market friction is formulated by coordination problems, and what is the policy to regulate the unemployment rate and monetary issues.

The early developments of directed search models raise the motivation from the basic issues of how the market is conducted when goods sold by sellers are indivisible and buyers are constraint by limited mobility. Peters [27] is among the earliest contributors to address the problem under such motivation. Using a complex utility form, he considered price competition among firms and established the form of the search game as a mixed strategy profile. His subsequent contributions included Peters [28], which proposed price posting as a matching technology and compared solution with the (ordinal) Nash bargaining solution; and Peters [29], which allowed for comparison among a subset of mechanisms and derived fixed price posting as an equilibrium in a dynamic decentralized market. Further, Peters [30] established market utility property as a simplified tool to directed search models in its limit form.

Moen [20]constructed a searching structure in a Diamond [9], Mortensen [21] and Pissarides [33] fashion, where buyers are price takers and firms post wages in a dynamic environment. He analyzed the equilibrium which he referred to as ”competitive search equilibrium”, and claimed that such equilibrium allocation is socially optimal. Burdett et al. [4] focused on equilibrium price and matching formulation, they simplified the framework of directed search models significantly with a basic linear payoff form. Through the structure of the sequential game, they showed that equilibrium friction comes from the coordination problem of buyers’ visiting game. They also allowed for heterogeneity among sellers in their capacities and evoked Lester [16]’s analysis of wage posting of multiple vacancy firms. This simple analysis has also been later on extended into dynamic environment to analyze the role of middlemen Introduction 4 in the economy by Watanabe [44], who endogenously rationalized the choice of inventory of sellers and therefore differentiated the middlemen from ordinary producers. 4 Meanwhile, Julien et al. [13] provided a bidding auction framework where sellers could compete by offering reserve prices instead of fixed prices, and realized the same urn-ball matching form in equilibrium. The bidding auction mechanism was supported by the later verification of Coles and Eeckhout [6], in which they proved that by using the setting in [4], the multi-equilibrium problem arises and this further implies that the mechanism proposed by Julien et al. [13] is indeed the optimum mechanism for sellers. Acemoglu and Shimer [1] provided a wage-posting search framework with risk averse works to derive the form of efficient unemployment insurance.

The recent developments of directed search models attempt to verify equilibrium properties from inﬁnite markets to check the robustness of such models. Among them, Galenianos and Kircher [10] summarized existing directed search literature allowing a generalized form of matching probability. They established the price posting structure as a sequential non-cooperative game, and provided that given certain restrictions on the matching probability form and focus on single price posting equilibrium, the unique equilibrium is indeed the one derived in [4]. Despite that, the restriction they took as given to prove the uniqueness of equilibrium was not yet desirable, it was inherited by Kim and Camera [14], in which they allows production heterogeneity on sellers side and provides the uniqueness of directed search equilibrium with a ﬁnite market.

The basic idea of directed search models is that buyers in the market observe the offer of each seller before deciding where to look for a deal that they find most desirable. Assumption of ”market utility property” was made in order to rationalize individual buyer’s visiting strategy as the trade-off between committing to a particular seller and his outside options. In equilibrium, visiting different stores endogenously yields an equivalent payoff.

4A earlier analysis in Watanabe [43] investigated inﬂation eﬀects of middlemen within a (S,s) price adjustment system, and a systematic analysis of role of middlemen has recently been provided in Watanabe [45] Introduction 5

Looking into this rationality and comparing it with those models of strategic bargaining, some interesting analogies show up. First, both of them consider payoff structures that yield mutual benefits among two parties, commonly referred to as the feature of ”cooperation” of bargaining models. Indeed, as I will illustrate in this thesis, directed search approaches not only apply to labor and monetary market, but also a more general group of cooperative behaviors. Second, in both of them, some threats are presented to facilitate a stable equilibrium. For example, strategic bargaining literature proposes that the bargaining process is agents putting forward alternative offers(See Harsanyi and Selten [11], Rubinstein [37] for example). Delays of reaching an agreement will discount benefits for both sides, playing a key role in the threat. While in directed search models, the threat simply takes the form of the probability that all buyers visit other sellers, having a similar effect to discounting seller’s benefit. Last but most obviously, the current literature of directed search models often derives (asymmetric) Nash bargaining solutions as equilibrium, supporting earlier conjectures of random search models.

How should we understand these analogies? In Chapter2, I revealed the link by proposing a Nash program within a two sided search environment. The Nash demand game I consider follows Anbarci and Sun [2] and Rubin- stein et al. [38], and endogenize the threatening probability with the matching probability contingent to buyers’ mixed strategy proﬁle of visiting. I show that price posting in frictional market can be viewed as a special case of Nash demand game with endogenous threatening probability; and therefore, the set of stable equilibrium, namely the asymmetric Nash solutions derived by strategic bargaining literature also apply to directed search models.

The beneﬁt from this comparison comes twofold. First, it gives a more general explanation about groups of cooperative behaviors within directed search environments without being restricted to the applications of labor and monetary economics. Such cooperative behaviors include co-authorship, school enrolment, marriage, etc. Second, it allows for the analysis of a large group Introduction 6 of non-linear payoﬀ structures and trading mechanisms with a directed search approach.

The analysis so far is based on the assumption that a seller could put forward a complete contract in the first round of the game. That is, sellers are able to put forward a contract which specifies all terms and conditions that could be realized upon matching. However, we learn from our daily observations in goods exchange markets that sellers usually produce, bearing a sunk cost before they enter the market, and the quality of the product is often contingent on the level of cost involved in the production. This particular situation, as in Rogerson [36], has been defined as externality in goods exchange market. Sellers often have to produce ex-ante because it is difficult for buyers to enforce the quality of goods if it is not yet produced when payment is made. With search friction, the level of ex-ante investment, the corresponding quality of the product and the equilibrium price with this technology have been interesting topics.

In Chapter3, I explore a directed search model where costly production pro- cesses should take place before sellers enter the market. Sellers produce the goods and bring them to the market, bearing their own cost, and the quality of the goods is contingent on the cost involved. I derive the equilibrium of quality level provided by the seller and the price level, and characterize comparative static properties. The major ﬁnding is that such production-in- advance constraint will generally derive second-best outcome which involves under-investment.

The analysis so far is based on the assumption that a seller’s posted offer is enforceable. Our daily observation presents a large frequency of trades (either in a physical form such as goods exchange or in a more implicit form such as cooperative behaviors) led under environments with externality; especially, as most of the goods traded in the market are produced in advance, and their quality is inevitably contingent on the producers’ costs it is often impractical Introduction 7 to assume the contract to be perfectly enforceable. In Chapter4 I exam- ine a match-contingent hold-up problem raised by externality and sequential investment. The payoff structure takes the form of a continuous prisoner’s dilemma game to capture externality. One party’s benefit is contingent to the other party’s costly efforts. The trading mechanism inherits Julien et al. [13]’s approach, so that in a pairwise match, the seller gets no benefit; while in a multiple match, the seller gets all the surplus.

The findings of Chapter4 come threefold. First, when contracts are totally unenforceable, that is, the seller must rely on local competition to get a benefit, it is generally not possible to implement the (Paretian) first-best allocation; and second-best allocation yields under-investment, which supports the commonly held view (See Hart and Moore [12] for an example). Second, in comparative statics analysis, I show that sellers tend to invest more if market tightness gets larger. This property looks a bit counter-intuitive, specifically because it conflicts with the fully-enforceable contract case. I explain this phenomenon with two channels, the extensive margin and the intensive margin effect. Fi- nally, when maintaining a payoff structure with externality and production in advance, the contract has to be assumed to be enforceable; that is, sellers are able to post a price that the buyer will commit to so the efficiency is improved, while the allocation is still under-invested. Chapter 2

A Cooperative Game with Coordination Friction

Suppose two parties search for partners to work on a project for mutual ben- eﬁt, where one party can commit to a contract which is contingent on the realized states of the world. When the other party chooses her partner, there are coordination frictions, as analyzed in Burdett et al. [4]. This paper exam- ines a generalized form of cooperative behavior with such a frictional searching framework, and shows that when the utility set is convex, the contract writ- ers will post contracts that are not contingent on the prevailing state of the world, though they could make use of these contingencies. Further, the unique equilibrium contract takes the form of a generalized Nash bargaining solution, where the relative bargaining power depends on the numbers of market participants.

JEL Classiﬁcation: C78, C72, D86. Keywords: directed search, bargaining problem, contract theory Chapter 2. Cooperative Game 9

2.1 Introduction

In daily life, we often observe situations where complex contracts are offered by competing parties. These contracts are usually state-contingent and often involve multi-dimensional elements which could affect the quality of the service/goods delivered in many ways. For example, consider the off-plan housing market. Before construction is complete, a development company usually specifies the price along with the design, materials in use, and other quality-relevant aspects of a house in advance. Perspective customers taking all these specifications into considerations, will make a decision from which company they wish to purchase.

The searching process of indivisible goods (such as house, cars, labor forces, etc.) often involves coordination frictions, as has been captured by directed search literature. However, in terms of the form of the contract, the scope of such literature is limited. The total surplus of each transaction is constant, and preference structures of both sides are often assumed to be linear. The model could not be used to study such cases as the pre-mentioned one that involves multi-dimensional action set. In this paper, I use directed search framework to study the formulation of such multi-dimensional contracts. Instead of ﬁxing the total surplus as constant, I allow the new non-linear dimension (cost) into the action set, and analyze a new form of equilibrium allocation. In the model, both the price and quality of the house could be endogenously determined by the market structure.

More generally, I investigate the matching process between two complementary parties (e.g. male and female) which could pairwise match and take cooperative actions. The complementarity among two parties lies in the fact that each of them could generate beneﬁt by matching with a single partner from the other party only, therefore my model captures most cooperative behavior that relies on pairwise partnership, such as indivisible goods exchange, mating, etc. The payoﬀ set I consider takes a generalized form which follows Chapter 2. Cooperative Game 10

Nash [24]1. I constructed the matching process in a directed search fashion: each male can commit to a contract, and then each female oﬀers one male she found most desirable to partner, on receiving more than one oﬀers, male choose one to partner with.

In an environment with uncertainty, the use of a contract is to assign to each possible state an action to take, as has been observed by Arrow [3] and Debreu [7]. In particular, the uncertainties involved between the moment the contract is posted and implemented enables a contract writer to write a state-contingent contract which assigns different actions to different states. Therefore, one of the first problems he encounters is to choose from such state-contingent contracts or a fixed value contract which assigns only one action to all states. In the context of directed search models, this problem is reflected by the long debate on selection of trading mechanisms almost since their inception.2 Intuitively, the equilibrium contract posted should be Pareto efficient so that no deviation is beneficial, which in turn, implies that if the set of utility is strictly convex, the male will post a contract that is not contingent to any uncertain state. Therefore, the risk aversion from either side of the market which provides a strict convex utility set, guarantees that male will post state- irrelevant contract. Yet, it is not a necessary condition because the action set itself can be strictly convex subject to some resource constraint, therefore even double-sided risk-neutral relationships could also provide convex utility. My results extend the existing models in that it generalized the payoff set to allow double-sided risk aversion, and my derivatives do not rely on ”market utility”

1The primitives of the action set in Nash [24]: compact, convex and contains a non-empty support. 2For example, Burdett et al. [4](hereafter BSW) takes fixed price posting as given while Julien et al. [13](hereafter JKK) takes bidding auction as the pricing mechanism when seller is visited by multiple buyers, and focus on competition on reserve price, which is the price when seller is met by single buyer. Later on, both encountered the challenge from multi-equilibrium issue addressed by Coles and Eeckhout [6], commonly referred to as ”indeterminacy” of directed search model. According to their analysis, by allowing sellers to post a list of match-contingent prices, the symmetric Nash equilibrium could be a fixed price as in BSW, reserve-auction combination as in JKK, but also, the combination of reserve price and any price that seller would charge when he is met by multiple buyers. Recently, Selcuk [40] showed that fixed price posting is the dominant trading mechanism when the buyer is risk averse, but his analysis use assumption of ”market utility” so that it only apply to large markets. Chapter 2. Cooperative Game 11 properties or homogeneity of agents so that it applies to finite markets and markets with heterogeneous agents.

The unique equilibrium contract I derived turns out to be a generalized Nash bargaining solution, where the relative bargaining power is a function of agents’ number from both sides of the market. Not surprisingly, males enjoy higher bargaining power when the competition from the female side is more intensive, and vice versa. In the last section, I showed the tractability of the model by solving several examples.

2.2 The Model

The set of participants separates into two groups, males M = {1, ...M} and females F = {1, ...F }.3 Both males and females are homogeneous among their groups, and partnered with one who belongs to the identical group does not generate proﬁt. Formally:

  0 if |S| = 0, 1 or > 2 t(S) = ,  0 if |S| = 2, and S ⊂ M or S ⊂ F where S ⊂ M ∪ F is a possible coalition of agents and t(·) denotes for the total value generated from the partnership.

A realized match is a pair of players {i, j} where i ∈ M and j ∈ F so that they can generate surplus by taking mutual actions. The feasible action set n 2 X ⊂ R :(x1, x2, ...xn) is compact and convex. The payoff function P ⊂ R : X → (V,U) gives a pair of utilities that male and female could obtain by taking the action in a realized match. Throughout the paper I assume P as a one-one reflection, that is, if two actions give the same payoff, they will be treated as identical. It is quite natural to impose the assumptions of (weakly)

3I use bold letters to denote sets and normal letters to denote corresponding elements. Chapter 2. Cooperative Game 12 risk averse on the payoﬀ function due to the nature of diminishing marginal productivity and utility. 4 This is formalized as:

Assumption 2.1. In a realized match, for any possibility q ∈ (0, 1) and 5 actions X1,X2,X where X = qX1 ⊕ (1 − q)X2 :

P (X) > qP (X1) + (1 − q) P (X2) (2.1)

The pure strategy action is at least as good as the corresponding mixed strategy for both male and female, which means they are both at least weakly risk averse.

Assumption 2.1 preserves the compactness and convexity of the utility set P from its domain X, which coincides with assumptions of Nash (1951). Fur- ther, for mathematical convenience, I assume P is bounded by a diﬀerentiable frontier P where the Pareto frontier PO : V¯ (U) ⊂ P (that is, the descend- ing segment of the upper part of P) is twice diﬀerentiable. Mathematically, ¯ 0 ¯ 00 V (U) < 0 and V (U) 6 0.

For a realized match, I focus on the form of commitment that is represented by an enforceable contract. Since there are uncertainty involved between the time contract is posted and implemented, it is natural to consider state-contingent contract which assign to each state a unique action to implement.

Though the states that a contract could be contingent on is fairly broad, there are two restrictions I shall impose on the space of such states. First, here I do not consider the situation when male could aﬀect the possibility of events after he posted the contract, the event should be irrelevant to males’ subsequent actions. In other words, I do not consider moral hazard issues. Second,

4In Nash [24], there is no role of risk aversion because he obtain convexity of the payoff set by the linear combinations of the pure strategy points on the boundary, the underlying reasoning being that a mixed strategy action gives exactly the payoff as the weighted average of its two sides. In here, I have a continuous boundary which is consisted by pure strategy actions, thus allow the analysis of risk aversion. 5 n Operation ⊕ take the mixed strategy of its two sides. Furthermore, for A, B ∈ R : A > B iff. Ai > Bi for all i; A > B iff. Ai > Bi for all i, and Ai > Bi for some i. Chapter 2. Cooperative Game 13 information is complete, so that both male and female know the probability of each state.

Denote Ω as the space of event that male could contingent his contract to, and l ∈ Ω is the element of the event space, state, the corresponding Z possibility of state l is denoted as ql , qldl = 1. Denoted the contract

Ω posted by male i as Ci, the state-contingent contract is a set of actions male i proposes that contingent to relevant state, which shows: Ci = {Xil}l∈Ω. Figure 1 shows the sequence of implementation of a state-contingent contract.

l ∈ Ω

{Xl} Xl

Figure 2.1: Sequence of Event

A signiﬁcant subset of the Ω, the meeting-contingent contracts, has been broadly used by directed search models. For example, in JKK and Coles and Eeckhout [6], they consider the posted contract contingent on the number of potential partner realized in the meeting. They assumed that, if the contract poster receives more than one partner, he has the opportunity to ex- ploit bidding and take the total surplus. This subset requires special attention because the probability of each state l ∈ (1, 2, ...N) now relies on buyers’ selection strategy. In here, to capture the most general set of contracts male could post, my setting of Ω includes the meeting-contingent contract, the meeting- irrelevant contract and all combination of them. For example, the following contracts from male are all allowed: ”depending on whether it rains or shines when we date, the time we spend on soccer is Xr or Xs”. Or, ”if it rains and

I receive one female, the time spend on soccer is X1r, otherwise it is X”, etc.

The matching process is a sequential contracting game between males and females: Chapter 2. Cooperative Game 14

1. Each male i ∈ M moves non-cooperatively, oﬀers a state-contingent

contract Ci = {Xil}l∈Ω.

2. Female j ∈ F observes posted contracts, and chooses one male with the M P mixed visiting strategy θj = (θj1, θj2...θjM ), where θji = 1. i=1 3. Each male chooses one female at random to form a partnership.

4. Depending on the state l, the contract is implemented as per the contract.

The equilibrium of this game consists the set of strategies that agents opti- mize at any particular stage, given the existing observations and anticipations of reactions of the rest of the market, as standard in directed search models, I ignore the equilibria that requires coordination among females in the visiting game, and focus on symmetric equilibria 6. Formally, it is deﬁned as the following:

Deﬁnition 2.2. The directed search equilibrium consists of: a set of contracts {Ci}i∈M , a mix strategy proﬁle {θj}j∈F, and a decision rule Γ :

{Ci}i∈M → {θj}j∈F, such that:

• Given the contract set {Ci}i∈M, {θj}j∈F is the symmetric (non-coordination) sub-game equilibrium of selection strategy;

• The decision rule Γ : {C } → θ∗ describes best response of i i∈M j j∈F visiting game to a contract set;

• Given the decision rule Γ, {Ci}i∈M is a sub-game perfect (Nash) equilibrium.

By this deﬁnition I ﬁnished the description of the model, and the rest of this paper attempt to derive the equilibrium of this model and discuss some properties.

6For a standard directed search setup, see BSW. Chapter 2. Cooperative Game 15

2.3 The Optimum Contract

Facing the state-contingent contracts, the ﬁrst problem a male encounters is what form of contract to post. This section endogenously derives males’ equilibrium contract choice and the properties under which the main theorem holds.

The following theorem shows that males will post ﬁxed value contract on the Pareto frontier which is irrelevant to any outside contingencies under a very relaxed condition.

∗ ∗ ¯ Theorem 2.3. For any male i ∈ M, he posts Ci = Xi ∈ V (U) for all l ∈ Ω iﬀ. V¯ 00(U) < 0.

Proof. See appendix.

The proof proceeds by contradiction. Suppose a male posts a state-contingent contract Ci = {Xil}l∈Ω, he will always find a profitable deviation to a fixed 0 ¯ value contract on the frontier Xi ∈ V (U), which maintains the female’s visiting strategy, but improves male’s payoff. Figure 2 illustrates this.

Xl1 0 Ci

Xl2

Figure 2.2: Optimum Contract

The competitive nature of males (contract posters) will make them post a contract on the Pareto frontier of the action set. Whenever this frontier is strictly concave, any state-contingent contract which yields different payoff in different realized state is inferior to some fixed value contract on the frontier. Chapter 2. Cooperative Game 16

Therefore, males will post fixed value contract which does not contingent on any outside states, though he could make use of such contingencies. This result in particular, is to argue that the ”indeterminacy” result discovered by Coles and Eeckhout [6] depends on the linear frontier, whenever the Pareto frontier is strictly concave, this indeterminacy disappears. Putting this result to a exchange economy context, not only the price sellers will post is fixed, but all the payment-relevant specification would be fixed for all states and written ex-ante by competitive sellers, I refer to this theorem as ”determinacy” of directed search models.

The following proposition shows conditions under which Theorem 2.3 holds.

Proposition 2.4. V¯ 00(U) < 0 iﬀ:

• (S1)At least one of male and female are strictly risk averse, or

• (S2)Support of P¯ is strictly convex.

Proof. Denote the support of the P¯ as X¯ . (S1): If at least one of male and female is strictly risk averse, the inequality (2.1) holds strict on every two ¯ ¯ ¯ ¯ points of X¯ . Pick any X1, X2 ∈ X¯ and q ∈ (0, 1), qX1 ⊕ (1 − q)X2 ∈ X due ¯ ¯ ¯ ¯ ¯ ¯ to convexity of X, then ∀P ∈ P, s.t.: P > P (qX1 ⊕ (1 − q)X2) > qP (X1) + ¯ ¯ ¯ (1 − q)X2. (S2): If P¯ is strictly convex, then for any X1, X2 ∈ X¯ , the convex ¯ ¯ ¯ ¯ combination q(X1, X2) = qX1 + (1 − q)X2 where q ∈ [0, 1] joint with X¯ only ¯ ¯ ¯ ¯ ¯ ¯ ¯ with X1, X2. Then, there exist X ∈ X s.t.: P (X) > P (qX1 + (1 − q)X2) > ¯ ¯ ¯ qP (X1) + (1 − q)P (X2).Given the twice diﬀerentiability of V (U), either of them suﬃciently implies V¯ 00(U) < 0.

Risk aversion from either side of the market guarantees the convexity of the frontier, therefore Theorem 1 applies to any (partial) risk aversion preference structure. This result generalized Selcuk [40]’s analysis which considers risk- averse buyers and focuses on meeting-contingent contracts, and shows that ﬁxed value contract is equilibrium choice of sellers. In here, I show that the Chapter 2. Cooperative Game 17

ﬁxed value contract as an equilibrium result, is robust to other trading mechanisms that could be contingent on meeting contingencies and non-meeting contingencies, and it applies not only to risk-aversion buyers, but any preference structure where at lease one of seller or buyer is risk-averse.

Yet, risk aversion is not necessary. As shown in the Proposition, paralleling to the condition of risk-aversion, the strict convexity of the action set could also guarantee a strict concave Pareto frontier, even if both male and female are risk neutral. For example, consider the linear payoﬀ structure which is double-sided risk neutral: P :X→ (V,U): V = x1,U = x2, and the action set 2 2 X subject to some resource constraint: X : X = (x1, x2) s.t. x1 + x2 6 1. Then P : {(V,U): V 2 + U 2 = 1} is convex, and satisﬁes V¯ 00(U) < 0. However, it does not associate to a risk averse preference structure.

The strict convexity of the action set implies that resources have strictly diminishing rate of marginal substitution, which is an widely accepted axiom in modern economic theories. The following equivalence of Proposition 2.4 shows the condition when the frontier V¯ (U) is linear.

Proposition 2.4’ V¯ 00(U) = 0 iﬀ :

• Both male and female are risk neutral, and

• X is linear segment, hyperplane or half-space.

Proof. This is just the duality of Proposition 2.4, so I omit the proof.

Proposition 2.4’ shows that the linear frontier which is usually considered in directed search models requires very strict conditions, actually, it is the only exception from a general set of convex payoff structure. Combining with Theorem 1 which states that linear payoff frontier is the reason which give rise to the indeterminacy, it is safe to conclude that fixed value contract (e.g. that is used in BSW) is the equilibrium choice of contract poster under most convex payoff structures, and does not rely on risk aversion preference structure. Chapter 2. Cooperative Game 18

The message conveyed by this section is simple, under reasonable conditions, the male (contract writer) will post a fixed value contract which does not contingent to any outside states. In terms of a fixed value contract, the one I consider here is a set of terms and conditions that contains not only price, but all payment-relevant specification that could be written in advance, such as quality, warranty, delivery, etc. Noticeably, my analysis does not rely on ”market utility properties”7which performs well only on large markets,or the assumption of homogeneity, therefore Theorem 2.3 applies to finite markets and markets with heterogeneous agents.

2.4 The Equilibrium

The analysis that follows maintains the strict convexity of utility set so that Theorem 2.3 is eﬀective. The main result from this section is that the unique symmetric equilibrium is a generalized Nash bargaining solution, where the relative bargaining power is associated to the market structure. I ﬁrst introduce the concept of generalized Nash bargaining solution within the context of my model, and characterize it with the relative bargaining power. Then I derive the unique equilibrium contract of the game, it turns out that the unique equilibrium contract is identical to a particular generalized Nash bargaining solution (GNBS), where the relative bargaining power is uniquely associated to the number of males and females in the market.

For any realized match {i, j}, the problem of contract C : {i, j} → (V,U) coincides with the allocation problem over two agents who take cooperative action, but there are alternative ways to collaborate, commonly referred to as the bargaining problem8. The well-known generalized Nash bargaining

7Peters (1990) gives a detailed analysis about this. 8The primitives of bargaining problem: two people, a set of feasible utilities, and a disagreement point as origin, where there are non-empty support of utility set. Osborne and Rubinstein [26] Chapter 2. Cooperative Game 19 solution for this problem ﬁrst introduced by Harsanyi and Selten [11] is deﬁned as follows9.

Deﬁnition 2.5. The α−Nash solution: Nα ∈ (V,U) solves:

∗ ∗ α Nα :(Vα ,Uα) = arg max {Υ = V U} hU,V i

¯ where V 6 V (U).

The concept of Nα was originally contrived to capture the asymmetric ability to negotiate for a more desirable payoﬀ. After simple algebraic manipulation, the following proposition characterizes Nα with α.

+ Proposition 2.6. For any α ∈ R , Nα :

• satisﬁes: V¯ 0 U 1 − U = . (2.2) V¯ α

• is well deﬁned;

∗ ∗ ¯ Proof. From the eﬃcient axiom of GNBS: (Vα ,Uα) ∈ V (U). To maximize Υ, take the f.o.c w.r.t U, it yields (2.2). The LHS of (2.2) increases with 10 + ∗ ∗ U, ranging from 0 (exclude) to ∞ , then for any α ∈ R ,(Vα ,Uα) is well deﬁned, this completes the proof.

+ With a relative bargaining power α ∈ R , equation (2.2) identiﬁes an Nα uniquely. It is clear that with larger α, male would have stronger position in the negotiation, therefore enjoys higher payoﬀ.

Now I derive the directed search equilibrium by the following proposition, as it turns out, the equilibrium contract that male would post take the same form with (2.2).

9My denotation of α is slightly diﬀerent from their paper, though the sets they indicate are identical. 10See appendix for detail; Chapter 2. Cooperative Game 20

Theorem 2.7. With settings in section 2, the directed search equilibrium is as such:

∗ ∗ ∗ ∗ ∗ • Males post identical ﬁxed value contract Ci = Xi , where Xi → (V ,U ) s.t: V¯ 0 U 1 − U = ; ¯ M M−1 −F M M V F ( M ) − F − M−1

F F [1−(1−θi1 ) ]Ui1 [1−(1−θi1 ) ]Ui2 • The decision rule Γ: = for every i1, i2 ∈ M F θi F θi X 1 2 that i1 6= i2, and θi = 1; i∈M • Females visit every male with same probability: θ∗ : θ∗ = 1 ; ji M j∈F,i∈M

• is well deﬁned.

From Theorem 2.7 and Proposition 2.6, the following Corollary comes imme- diately:

Corollary 2.8. The unique symmetric equilibrium contract posted by males M M−1 −F M M is an Nα where α = F ( M ) − F − M−1

This corollary incorporates the equilibrium contract of directed search model with a speciﬁc (asymmetric) Nash bargaining solution. The relative bargaining power is associated to the number of males and females. It is easy to see that α is increasing with N, but decreases in M, which implies that males enjoys larger bargaining power when there is more females, but suﬀers when there is more males.

2.5 Examples

In this section, I close the paper by providing applications of the models pre- mentioned in the introduction.

Goods Exchange Chapter 2. Cooperative Game 21

Consider the goods exchange market with M sellers and F buyers. Suppose each seller could produce one indivisible product, and the quality of the product depends on the cost involved c. The production function takes the

1 Cob-Douglas form f(c) = c 2 . On receiving the amount of money p, the risk

1 averse seller receives utility: u(p) = p 2 . With directed search framework, sellers post the contract (p, c), and buyers visit. When they exchange, buyer

1 1 gets: U = c 2 − p, and seller gets: V = p 2 − c

Using Corollary 2.8 and also according to envelop theorem, the equilibrium contract (p∗, c∗) solves:

∗ ∗ n 1 α 1 o (p , c ) = arg max Υ = (p 2 − c) (c 2 − p) hp,ci where α is the one presented in Theorem 2.3.

Take the f.o.cs:

∂Υ α − 1 1 α−1 1 1 α = 0 : p 2 (p 2 − c) (c 2 − p) = (p 2 − c) ∂p 2 − 1 ∂Υ 1 α−1 1 c 2 1 α = 0 : α(p 2 − c) (c 2 − p) = (p 2 − c) ∂c 2

Which simpliﬁes to:

1 1 1 1 pc = ; 2αc 2 (c 2 − p) = (p 2 − c); 16 it solves:

2 2 ! 1 2α + 1 3 1 α + 2 3 (p∗, c∗) = , 4 α + 2 4 2α + 1 Chapter 2. Cooperative Game 22

For comparative static analysis, when there are more buyers searching for goods, sellers would enjoy stronger bargaining power, therefore charge a higher price but provide lower quality in equilibrium.

Battle of the Sexes

My model could capture more complicated form of matching that features strategic complementarity cooperative relationships. Consider the following version of ”Battle of the Sexes” game11 within a searching framework: males prefer watching soccer while females prefer watching ballet, though both value spending time together more than doing anything separately. Moreover, male’s utility of watching soccer is indirectly eﬀected by the female’s mood, which is related to the time they spend on watching ballet, and likewise, female’s utility also relies on male’s mood. Suppose males attempt to approach females by their proposal of time they would spend on soccer or ballet, what is the equilibrium proposal?

The use of this generalization is fairly clear. In reality, many cooperative behaviors could not be represented by trade offs of direct side payments. Rather, more complex interactions between partners are often observed. For example, consider the case of commercial land resource exploitation. Governments care about social gain while capital owner cares about profit of the factory. The social gain is the product of the labor intensity and the profit (due to its effect to the salaries), and the factory’s profit depends on the intensity of the capital and the political support from the government, which is a function of government’s social gain. My model provides micro foundations of the partnership formulation, and helps understanding equilibrium activities of such cooperative behavior from a market competitiveness prospect.

11The game I described below is actually ”Hawk and Dove” in classical game theory. The only difference between the two game is that ”Battle of Sexes” emphasis on the coordination between players while ”Hawk and Dove” captures competition over the shared resources. However, if we use Nash [24]’s definition of the payoff set and make every action between two pure strategy profile feasible by taking mixed strategies, then both games yield the same payoff set. I hereby abuse the term to emphasis its implication in the process of matching. Chapter 2. Cooperative Game 23

Consider the mating market with M males and F females. Suppose the time spent on ballet does not generate utility for male, and on soccer does not generate utility for female either. However, when watching soccer, male have to be accompanied by female, and utility of watching the soccer is eﬀected by female’s mood, which writes:

3 1 V = s 2 U 2 , (2.3) where s is the time they spend on football, and U is utility of female. Likewise, the female gets the utility from spending time on ballet, which is also aﬀected by male’s mood: 3 1 U = b 2 V 2 , (2.4) where b is the time they spend on ballet. Each of them has a mass of 1 unit of time to spend together, so the time constraint: s + b = 1. Now in the local market, males approach females with their proposal (s, b) and females choose one to date.

We ﬁrst derive the forms of V,U in terms of s and b. Substituting (2.4) to (2.3):

3 1 3 3 1 2 V = s 2 U 2 = s 2 b 4 V 4 ⇒ V = s b;

Similarly: U = sb2. This payoﬀ structure can be shown in Figure 3.

Vmax

OUUmax

Figure 2.3: BoS Game Chapter 2. Cooperative Game 24

It is straight forward to verify the convexity of the set P which guarantees the uniqueness of the equilibrium. According to Corollary 2.8, the equilibrium proposal (s∗, b∗) solves the following:

(s∗, b∗) = arg max Υ = (s2b)α(sb2) hs∗,b∗i s.t : s + b = 1

To solve this, construct lagrangian:

L = Υ + λ(1 − s − b) f.o.cs:

∂L = 0 : (2α + 1) s2αbα+2 = λ ∂s ∂L = 0 : (α + 2)s2α+1bα+1 = λ ∂b ∂L = 0 : s + b = 1 ∂λ

Which solves: 2α + 1 α + 2 (s∗, b∗) = , 3α + 3 3α + 3

When α = 1, male and female enjoys symmetric bargaining power, therefore they split time equally. α increases with the number of females, which indicates that competition among females will enable males proposing a larger proportion of soccer time, otherwise, they have to sacriﬁce more time to spend on ballet. Chapter 2. Cooperative Game 25

2.6 Conclusion

This paper examined the searching process of a bilateral partnership that generates cooperative payoffs. The searching process features the sequential game structure of directed search models, but the cooperative structure I consider which includes non-linear exchange structure and strategic complementary relationships was not captured by existing directed search models. My model provides micro foundation to the formulation partnerships that could not be analyzed in a the exchange structure, and thus helps understanding equilibrium cooperative behavior of such partnerships from a frictional market prospect. The major finding comes twofold. First, under reasonable condition, it is shown that equilibrium contract is state-irrelevant. This resolves the long debate of trading mechanism selection among directed search models. Generally, in an exchange market, sellers would prefer to post fixed price and provide fixed quality regardless of any outside state to reduce the uncertainty involves, even though they could make use of such contingencies. Second, the equilibrium of the mode takes the form of a generalized Nash bargaining solution where the relative bargaining power of each party is associate to the number of participants in the market. Chapter 3

Pricing and Quality Provision of a Frictional Market

I explored a directed search model where costly production process should take place before sellers enter the market. A seller produces the goods and brings them to the market, bearing his own cost, and the quality of the goods is contingent to the cost involved. I derived the equilibrium of investment level provided by the seller and the price level, and characterized comparative static properties. The major ﬁnding is that prior production will generally lead to second-best outcome which involves under-investment.

JEL Classiﬁcation: D83, J64, M37. Keywords:Contract theory, Directed Search. Chapter 3. Quality Provision 27

3.1 Introduction

In its most general form, directed search models capture situations where contracts are enforceable and payoffs are usually linear. 1 However, we learn from our daily observations in goods exchange market that sellers usually produce before they enter the market, bearing a sunk cost, and the quality of the product is often contingent on the level of cost involved during production. This particular situation, as in Rogerson [36], has been defined as externality in the exchange market. Sellers often have to produce ex-ante because it is difficult for buyers to enforce the quality of goods if they are not yet produced when payment is made. With search friction, what the level of ex-ante investment is in accordance with the corresponding quality of the product, and how the equilibrium price is determined with this technology have been interesting topics.

This chapter studies hold-up problems raised by this externality and production in advance within a frictional market. Costly investment on the sellers’ side benefits the potential buyer, and this captures externality in the goods exchange market. I construct a model of a two sided market where sellers produce in advance, and buyers visit sellers according to the quality that the seller provides and the price that the seller will charge. Sellers (the first mover of the game) compete with each other by offering different qualities of products, along with different price levels to direct the search.

My model extends the goods exchange models which apply price posting as the trading mechanism, and consider the searching process as a micro-founded non-cooperative game, commonly known as directed search models. Standard directed search models mostly focus on the case where sellers’ posted contracts are fully enforceable, that is, sellers can specify and realize every contingent that is posted ex-ante. Results of my model apply to most goods exchange markets with friction; in particular, the recent development of monetary search

1See for examples, Peters [28], Moen [20], Burdett et al. [4], and in the model in Chapter2 provided a general extension of this group Chapter 3. Quality Provision 28 models such as Trejos and Wright [42] and Lagos and Wright [15] belongs to this group. The current literature of monetary models usually assumes that trading be led under enforceable contract and Nash bargaining solutions be derived as equilibrium allocation. Our suggestion is that with production in advance, such characterization is not perfectly realistic, and at least a proportion of under-investment allocation, which derives from the model I present here, should be taken into consideration.

3.1.1 Related Research

My research is related to the strand of premarital investment game models and the prior production models. Among them, Peters [31] investigated premarital investment with a random search framework, in which the payoff structure is much similar to what I am presenting here: each child’s premarital investment is contingent to his/her parents’ costly investment, but generates utility to his/her partner only. They showed that though the nature of matching function could partially internalize the externality of the premarital investment, the equilibrium outcome however, though not surprising, is generally inefficient. Later on, the premarital investment game has been formally established by Peters and Siow [32], in which he endogenized the choice premarital investment by using a competitive approach and derived the assortative form of equilibrium. It shows that mixed Nash equilibrium converges to degenerate pure strategy equilibria which involves over-investment from both sides of the market. Furthermore, Masters [18] focus on monetary implications such as inflation and welfare effect of the prior production. Aside from the similar inefficient conclusions, he also verifies the validity of Friedman rule and suggest a moderate level of inflation.

My model diﬀers from this strand of models signiﬁcantly in its focus and approach. Unlike the general form of the matching function used in these models, I focus on individual sellers’ rationality of investment intensity in a Chapter 3. Quality Provision 29 directed search framework. I start my analyze from small market the equilibrium matching function is endogenously derived from individual’s rationality. Instead of focus on macro implications of the prior investment models, I re- late my model to contract theory, and verify that the commonly held view of under-investment under imperfect contractual environment still holds, with a solid micro foundation, in a directed search framework.

3.2 The Model

The market is populated by the set of S = {1, ...M} sellers and B = {1, ...N} buyers. Both buyers and sellers are homogeneous among their groups, and denote the buyer-seller ratio Φ = N/M as market tightness.

Each player is capable of producing one non-storable good by making a one-shot investment decision c. The good could be consumed by the buyers only, and the quality of the good is increasing with the investment level. This could be represented by buyer’s utility function u(c), where u0(·) > 0.

The searching process is a sequential game between buyers and sellers, and it consists of the process of each seller and buyer making a decision of investment level to produce a good and search for a partner to sell it:

1. All sellers make an investment c to produce a good before entering the market, and post the price p

2. All buyers observe qualities of the goods and price, and commit to P purchase from seller j with probability θj, where θj = 1.

3. The seller deal with the buyer who commits to the purchase; if multiple buyers wish to buy from one seller, he randomly chooses one to deal with;

4. All payoﬀs realized. Chapter 3. Quality Provision 30

For a realized deal, the seller receives V = p − c, and the buyer receives U = u(c) − p, and the production function takes the standard assumptions: 0 00 0 u (0) > 1 , u (·) 6 0 and u (∞) < 1 , which guarantees the finite total surplus. We define the social optimum choice of investment asc ¯. It yields that u0(¯c) = 1 and is well defined. This payoff form captures the idea of externality. If one seller produces, but there are no buyers who wish to purchase, then he bears a dis-utility −c. All buyers who do not purchase yield 0 benefit.

I ﬁrst derive the symmetric equilibria in a ﬁnite market, and then extend to large markets in this section. I use backward induction to derive agents’ optimum action in each stage, and I proceed from the simplest case of N = 2 and M = 2 and then extend to more general cases.

3.2.1 The 2 × 2 Case

I shall start from the buyer’s visiting strategy at stage two on observing sellers’ effort vectors in the market c = (c1, c2). Denote Uj as the benefit that the buyer will get from committing to seller j, and consider buyer i’s visiting strategy, where i, j = 1, 2. If buyer 1 is the only one who visits seller 12, then he gets the product for sure. Otherwise, if both buyers visit the same seller 1, 1 then the probability of them getting it is 2 . For any given seller’s effort level c , this yields: 1   u(c1) − p1, b1 = 1 U1 = 1  2 [u(c1) − p1], b1 = 2

Let Uij denote the (ex-ante) expected utility that buyer i visits seller j, and

θi denotes for the probability that buyer i visits seller 1:

1 U = (1 − θ )[u(c ) − p ] 11 2 2 1 1 2Without loss of generality, through most of this chapter, I consider interaction between buyer 1 and seller 1 as an example, situations of buyer 2 and seller 2 are symmetric. Chapter 3. Quality Provision 31

1 Similarly, U12 = 2 [1 + θ2][u(c2) − p2], buyer 2’s payoﬀs are symmetric. I seek mixed strategy Nash equilibrium in this sub-game. For buyer 1, he solves the problem of maximizing expected utility, which yields:

maxU1(θ1 | θ2) = θ1U11 + (1 − θ1)U12 hθ1i 1 1 = θ (1 − θ )[u(c ) − p ] + (1 − θ ) (1 + θ )[u(c ) − p ] 1 2 2 1 1 1 2 2 2 2

And buyer 2’s decision rule yields symmetry. Following standard, I focus on a symmetric mixed strategy Nash equilibrium, which yields:

2 [u(c1) − p1] − [u(c2) − p2] ∗ θ1 = θ2 = ≡ θ [u(c1) − p1] + [u(c2) − p2]

∗ Taking buyers’ mixed strategy equilibrium proﬁle θ1 = θ2 ≡ θ in stage two as given, I consider the sellers’ producing strategy in stage 1. Seller j’s payoﬀ depends on the number of buyers who commit to his program, denoted as bj.

Then it shows:   −cj, if bj = 0 Vj = .  pj − cj, if bj = 1, 2

Only no one visits seller j will he get a negative payoﬀ, otherwise he will receive pj − cj. Conditional on seller 2’s production with an eﬀort level x2, seller 1 solves the problem:

∗ 2 max V1(c1, p1 | c2, p2) = −c1 + 1 − (1 − θ ) p1 (3.1) hc1,p1i

Seller 2’s problem yields symmetry. First order conditions of the sellers yields: Chapter 3. Quality Provision 32

∂V1 = 0 : ∂c1 0 6 {2 [u(c2) − p2] − [u(c1) − p1]} [u(c2) − p2] u (c1)p1 = 3 {[u(c1) − p1] + [u(c2) − p2]} ∂V1 = 0 : ∂p1 6p1{2[u(c2)−p2]−[u(c1)−p1]}[u(c2)−p2] = {2[u(c1)−p1]−[u(c2)−p2]}{4[u(c1)−p1]+[u(c2)−p2]}

{[u(c1) − p1] + [u(c2) − p2]}

Although situations of the asymmetric equilibria of our model are potentially interesting topics, coordination problems in large markets get involved that are no less than that of the visiting game of second stage buyers. Suppose

(cj1 , pj1 ) and (cj2 , pj2 ) are equilibrium action pairs of two homogeneous sellers, where cj1 6= cj2 , to implement these separately, they need to coordinate base on equilibrium choice of each other, which is usually impractical; especially in large markets. We therefore focus on the symmetric equilibriums of sellers’ action through out this chapter. By the above proposition, we are able to establish the following proposition:

Proposition 3.1. The symmetric Nash equilibrium action pair of investment and posted price in stage one (c∗, p∗):

0 ∗ 4 4 • it is well deﬁned and u (c ) = 3 iﬀ. u(0) > 3

∗ 1 ∗ • p = 2 u(c )

0 4 Proof. The condition u (0) > 3 is to guarantee the existence of equilibria. We follow the track that ﬁrst proves the assigned point of c∗ belongs to the group of symmetric Nash Equilibrium, and then proved its uniqueness.

3.2.2 The N × M Case

We extend the existing model to the case the market consists of N buyers and M sellers. As discussed above, we focus on symmetric equilibria where Chapter 3. Quality Provision 33 all buyers adopt the same mixed strategy. Like before, we first figure out the buyer’s sub-game equilibrium upon observing utilities offered by sellers u =

(u(cj) − pj), and look for the Nash equilibrium of the actions that the seller would oﬀer.

Upon observation of the seller’s announcement vector (u(cj)−pj), an arbitrary buyer i visits seller j with possibility θij according to his expected utility of doing so:

Uij = Ωij [u(xj) − pj]

where Ωj denotes the probability that he will actually realize that the deal with the seller is conditional on buyer i visiting j. Clearly, following the results from chapter 1, the expression of the probability shows:

N (1 − (1 − θj) ) Ωj = Nθj

For each individual buyer i, he opts his mixed strategy to maximize the expected payoﬀ, the problem yields:

( M ) X max Ui = θijΩij [u(xj) − pj] hθj i j=1 M X s.t. θj = 1 j=1

Let L denotes the Lagrangian:

M ! M X X L = θijΩij [u(xj) − pj] + λ(1 − θij) j=1 j=1

F.O.C. yields: Chapter 3. Quality Provision 34

∂L = 0 : [u(xj) − pj]Ωij = λ ∂θj M ∂L X = 0 : θ = 1 ∂λ j j=1

The seller’s ﬁrst stage game is conditional on the buyers’ equilibrium visiting strategy. Seller j’s problem given other sellers’ choice U¯ −j yields:

N max E V (uj, cj | u−j, c−j) = −cj + (1 − (1 − θj) )pj M X s.t ΩjUj = Ω−jU−j = λ, and θj = 1 j=1

The solution to this problem is described by the following Proposition:

Proposition 3.2. The symmetric Nash equilibrium action pair of investment and posted price in stage one (c∗, p∗):

∗ 0 ∗ 1 • c is well deﬁned and u (c ) = M−1 N 1−( M ) ∗ ∗ 1 • p = u(c ) 1 − M M−1 −N M 1 N ( M ) − N − M−1

Proof. See appendix.

3.2.3 Large Market

In this section, I consider ”large markets”, in which both N and M is beyond certain boundaries and mathematically could be described as N,M → ∞, but Φ < ∞. As before, I focus on symmetric equilibria in which all sellers posting the same x and buyers using the same mixed strategy θi(x). The mixed strategy is directed by sellers’ vector x, on and oﬀ-equilibrium path. Chapter 3. Quality Provision 35

The mixed strategy generates a binomial distribution of buyers over sellers, and hence, an endogenous matching function. The probability that a seller M n N−n will be selected by n buyers given is Pr{n|x} = n (θi(x)) (1 − θi(x)) . In any symmetric equilibrium, since all sellers will also post the same x, then 1 θi(x) = M , ∀i ∈ M, when M,N go to inﬁnity with a constant ratio Φ, the Φne−Φ matching rate shows: limN,M→∞ Pr{n|x} = n! . Sellers face a Poisson ar- PN Φne−Φ −Φ rival of buyers. Since n=0 Pr{n|x} = n! = 1, hence Pr{n = 0|x} = e , and Pr{n = 1|x} = Φe−Φ. It is well known that directed search model taken to the limit will generate the same outcome as a competitive search, so I simplify the analysis and focus on the competitive search construct, but with a Poisson matching technology.

I consider one seller deviating to (ˆc, pˆ), while all other sellers provide (c, p). The program for a deviating seller is

n o V = max −cˆ + (1 − e−φˆ)ˆp (3.2) ˆ

(1 − e−φˆ) s.t.Uˆ = [u(ˆc) − pˆ] ≥ U¯ (3.3) φˆ ˆ ˆ where φ = φ(ˆx, x−1) 6= Φ is the expected ”queue length” generated by the deviation (ˆc, pˆ) 6= (c, p). The utility level U¯ is the maximum expected utility a buyer can obtain from any other sellers in the market. In the competitive search literature, this is referred to as the market utility property. A deviating seller takes this maximum utility as given, the exact reason why this construct is competitive.3 A deviation by a seller, creates an adjustment in the expected queue length, which operates much like a standard demand function following a price reduction.

The following proposition extends the previous model to its limit form:

Proposition 3.3. In a market where N,M → ∞, the symmetric Nash equilibrium action in stage one (ˆc, pˆ):

3Notice that this program diﬀers slightly from the competitive search construct. Here we could solve for y from the constraint, substitute it back into the objective function and maximize choosing x and Φ. Chapter 3. Quality Provision 36

∗ 0 ∗ 1 0 1 • c is well deﬁned and u (c ) = 1−e−Φ iﬀ. u (0) > 1−e−Φ

∗ eΦ−Φ−1 ∗ • p = eΦ−1 u(c )

Proof. Taking Proposition 3.2 to its limit form, this proposition yields straight.

This proposition characterized the equilibrium quality provision and pricing rule in large markets. The first part describes the equilibrium quality provided by a seller in the presence of the hold-up problem. Sellers are discouraged to invest in advance unless the market tightness is sufficiently high, and they will have enough probability to sell the product at the desirable price. For a particular payoff structure u(·), there exists a ”benchmark” market tightness contingent on u0(0) ,so that under this market tightness sellers will not start an h i ˆ 1 ˆ investment. Explicitly, Φ = − ln 1 − u0(0) where Φ denotes the least market tightness that enables sellers to have a positive expected value to enter the market. The second part establishes the relationship between equilibrium price and the investment level given the market structure. In general, the competition among buyers motivates sellers to input a higher investment level and provide higher quality, along with higher probability that the good will be purchased, this will enable seller to charge a higher price in equilibrium.

3.3 Characterization

This section attempts to characterize the equilibrium. On the ”intensive margin”, I characterize equilibrium action pairs (c∗, p∗) in aspects of comparative statics and eﬃcient properties. I also characterize the properties on ”extensive margins” such as equilibrium matches and free entry. Chapter 3. Quality Provision 37

3.3.1 Comparative Statics

Firstly, I evaluate whether the equilibrium c∗ is increasing or decreasing in Φ, this is illustrated by the following proposition.

Proposition 3.4. The equilibrium eﬀort c∗ is increasing in market tightness, and it converges to c¯ when market tightness goes to inﬁnity:

dc∗ > 0 and c∗(Φ) | =c ¯ dΦ Φ→∞

1 Proof. As u(·) is strictly increasing and 1−e−Φ is strictly increasing with Φ, this proposition comes straight.

I verified that with higher monopoly power, sellers are willing to provide a higher level of investment, and it converges to a social optimum level when market tightness goes to infinity. The absolute monopoly market is the market structure most likely to be efficient in this frictional environment.

3.3.2 Matching

The matching form follows what has been derived in standard directed search models, commonly known as ”urn-ball matching” technology. The expected number of successful buyer-seller matches, which is known as ”matching function” yields:

1 Γ∗(N,M) = M 1 − (1 − )N M

3.3.3 Free Entry

My model also allows endogenous equilibrium market tightness via free entry constraint. It is difficult to derive the explicit numerical form in a finite market, however, since the logic of an infinite market is very similar to the Chapter 3. Quality Provision 38

finite market case, and could provide approximation that is unbiased, efficient and consistent, so I focus on the case of an infinite market. Suppose the cost ¯ ¯ of entry is endogenously given by k > 0. Clearly, if k = 0, then the free entry market tightness Φ(¯ k¯ = 0) is the least market tightness a firm will choose to enter the market with:

1 Φ(¯ k¯ = 0) = Φˆ = − ln 1 − u0(0)

Suppose now that every ﬁrm has to pay a positive entry cost k¯ to enter the market. The equilibrium entry satisﬁes the following condition:

Φ¯ ∗(k¯) : (1 − e−Φ¯ )p∗ − c∗ = k¯ (3.4)

Both c∗ and p∗ are a functions of Φ¯ ∗, the RHS is a sole parameter function of Φ,¯ this equation gives equilibrium entry Φ¯ ∗. Free entry properties of our model can be illustrated by the following proposition:

Proposition 3.5. For any entry cost k¯ ∈ [0, u(¯c) − c¯), the market tightness Φ¯ ∗(k¯) which satisﬁes 4.8:

• is well deﬁned.

dΦ¯ ∗ ¯ ∗ ¯ • dk¯ > 0 and Φ (k) |k¯→c¯→ ∞

Proof. See appendix for details.

The free entry condition shows the benchmark market tightness that the seller should enter the market, and depicts the entry cost of the market. According to this condition, the seller should choose to enter the market or not according to the entry cost incurs and the market tightness. There are also straight policy implications. In reality large proportion of the entry cost could be regulated by governments, such cost includes investment tax, licensing fee, Chapter 3. Quality Provision 39 etc. Given a free entry condition, governments could adjust the industrial structure by regulating the relative entry costs throughout diﬀerent markets, therefore optimizing the eﬃciency of the economy.

3.4 Application

Goods Exchange Market with Monetary Transfer

My model has a standard goods market application with a costly production ex-ante. Consider the basic environment of Burdett et al. [4]: Each seller wants to sell a unit of product which costs 0 to produce, but generates 1 unit of utility for the buyer. If the buyer pays the seller the price of p, he will receive the utility U = 1 − p, and the seller gets a surplus of V = p. It is clear that this set up does not fit into our framework perfectly because of uniform utility. However, we can provide a modification to their set up to fit into our framework. Consider buyer’s value function: u(c) = 1 − e−ac, then the net benefit: U = 1 − e−αc − p. and sellers’ value function: V = p − c.

Clearly, to satisfy my model’s settings, α > 1 should be assumed, and the ln α eﬃcient investment:c ¯ = − a

Using our model, given market tightness Φ, the investment level c∗ satisﬁes:

1 αe−ac = 1 − e−Φ ln α(1 − e−Φ) ⇒ c∗ = α

And the equilibrium price: Chapter 3. Quality Provision 40

eΦ − Φ − 1 p∗ = u(c∗) eΦ − 1 eΦ − Φ − 1 α eΦ − 1 − 1 = α (eΦ − 1)2

The free entry condition must be satisﬁed:

1 Φ > Φˆ = − ln(1 − ) α

3.5 Concluding Remarks

I provided a general framework to analyze hold-ups when production must be taken before sellers enter the market. The general intuition is that this leads to second-best equilibriums that involve under-investment.

In the presence of a positive probability of hold-up situations which yield negative payoﬀs, sellers have been discouraged to enter the market until market tightness reaches a level that bears a positive economic proﬁt. Indeed, this benchmark market tightness is contingents on a particular production technology. I provided the benchmark of the market tightness given the production form.

Comparative statics show that sellers tend to provide higher quality products if market tightness goes higher. On the other hand, the investment level is lower than any of the pre-mentioned bargaining solution contracts, which supports the commonly held view that existence of hold-up situations will lead to under-investment. Free entry properties show that higher capital investment barriers will require higher market tightness in equilibrium, which will enable governments to regulate industrial structure by adjusting pre-investment barriers among industries. Chapter 3. Quality Provision 41

My model applies to a proportion of goods market exchanges with monetary transfer, as in the application section, we provided an extension of Burdett et al. [4] and illustrated some numerical properties. It explains for the situation where production should be made in advance due to technology limitation or lack of trust, and goods are non-storable. I believe it is plausible to assume such proportions exist, at least to some extent, in any form of monetary economy. From a more macro based view, how the economy would perform in the presence of such hold-ups, instead of assuming a exogenous bargaining power for all markets, could be an interesting topic. Chapter 4

Competitive Cooperation

I construct a directed search model where agents’ actions are pure externality in a non-enforceable contractual environment. Externality is captured by assuming one party’s costly production can be only consumed by the other party. The trading process follows the paradigm of directed search models, so that both parties take actions sequentially and seek for partners. Without an enforceable contract, the seller directs the search with costly effort, which will transfer utility to his potential partner. The benefit that the seller will get is contingent on the number of buyers who offer to partner with him. If there are no buyers or only one who shows up, then the seller will get zero profit, and the cost is sunk. Only when there are multiple buyers competing for one seller can the seller reap the benefit. I derived the unique symmetric equilibrium, which supports the commonly held view that imperfect contracts lead to second-best equilibrium, which involves under-investment.

JEL Classiﬁcation: D83, J64, M37. Keywords:Contract theory, Directed Search. Chapter 4. Competitive Cooperation 43

4.1 Introduction

Directed search models share assumptions of three dimensions to capture frictions in the market: capacity constraint, limited mobility and lack of coordination among agents. In terms of the form of the contract, they normally consider enforceable contract situations such as a linear payoﬀ transfer or a bargaining frontier, and derived bargaining solutions in equilibrium. While a large frequency of trades (either in its physical form such as goods exchanges or in a more implicit form such as cooperative behavior) are led under incomplete contractual environments, they hardly appear in directed search literature. In fact, when the ex-ante posted contract to ”direct” the search is completely unenforceable, it becomes diﬃcult to model agents’ searching strategies and subsequent behavior.

By its nature, cooperative behavior often involves externality. The term ”cooperation” stands for the situation where different parties input their efforts on a common project out of mutual interest, which in itself implies some extend of externality. Likewise, the term ”trade” which is defined as the situation where two complementary parties exchange products or services to potentially achieve a Pareto improvement solution, inevitably get externality involved1. A common observation in our daily life shows that many kinds of trading get externality involved. For example, in goods market, the quality of a product that a seller sells to a buyer is often contingent on the seller’s ex-ante costly investment level. In the case of schooling, a professor’s knowledge level and reputation usually benefits his potential students. In a marriage market, a girl’s dowry benefits the man she wishes to marry, and is certainly costly for her own family. Without taking externality into consideration, one could hardly understand many forms of economic activities. That is why since Coase [5], externality and corresponding contractual problems give rises to a huge

1The equivalence of trading and cooperative behavior in economic terms has been raised by Nash [23], and later systematically studied by cooperative game literature, see for example, Shapley and Shubik [41] Chapter 4. Competitive Cooperation 44 branch of contract theories, and become an important aspect in economic theories.

On the other hand, many contracts are made without a fully enforceable environment under many sources of constraints, rather, possibility of renegotiation exists to a considerable extent. One interesting scenario that comes to mind is the last scene of the movie ”Inglorious Bastards”: after Christoph Waltz made a deal with the general and action was taken to stop the war; Brad Pitt as the executor of the deal, chose to renege from the terms and killed Christoph’s assistant, Hermann on the way back. This is a typical situation of a hold-up problem in our daily life, especially with cooperative behavior. After action is taken from one party, and the benefits are partially realized, the other party is often given a chance to renegotiate as long as punishment of doing so does not overweigh the benefit. In fact, as such said by Brad Pitt: ”Most likely chew out me, I have been chewed out before.” means that the punishment is very slight in comparison to the benefits: ”to revenge as many Nazi mem- bers as possible”. In an ongoing cooperative relationship (as for Brad and the general in the movie) where actions are taken sequentially, punishments are often insufficient since a severe punishment will stop further mutual benefits from happening, therefore renegotiation is more likely to happen in such circumstances.

This chapter studies a matching model when payoff structure involves externality and contractual environment is not fully enforceable. I construct a model of a two-sided market where sellers and buyers take actions sequentially to search for partners. In terms of payoff structure, I consider costly efforts on both sides benefit the their partners to capture externality. Sellers (the first mover of the game) compete with each other by offering different effort levels in the first round, will direct the search. Buyers observing the benefits they can get from sellers, will compete by offering their own costly efforts, which benefit the sellers back. This offering process being the second round. In the Chapter 4. Competitive Cooperation 45

final round, the seller chooses the buyer with the highest level of effort and payoff realized.

My research is closely related to Julien et al. [13], who consider searching process as a micro-founded non-cooperative game. I realize the designs of game structure in directed search models which often have complete contractual environment neglecting an important fact: with the observation of the number of their partners, limited mobility of buyers is not suﬃcient to prevent renegotiation from happening, and the proposed contract is no longer enforceable without an explicit punishment device. In particular, after each buyer commits to a deal with one particular seller, if the seller observes that there is more than one buyer that shows up to his program, then he will hold up the property and extract the surplus. Likewise, if a buyer observes that he is the only one who is committed to a particular seller, then he has the incentive to ”hold up” the seller. Either way, the ﬁrst round contract is not credible.

My model applies to the market of the pairwise relationship forming where there is no point of access to make the ex-ante contract enforceable. The particular example I have in mind is the searching of co-authorship. By its nature, such cooperative behavior get externality involved, and it is not often practical to introduce money transfer in such a relationship. The general intuition is that with the matching technology, there is probability of multiple matches as well as a pairwise match, and the conﬂiction between capacity constraint and multiple local candidates gives rises to local competition. In equilibrium, this competition alleviates the prisoners dilemma. Therefore, I refer to our model as ”competitive cooperation”.

Results of our model also apply to many forms of frictional markets including monetary exchange models and labor search models. Back from Diamond [8], Mortensen [21] and Pissarides [34], search theory has a long history of incorpo- rating a perfect contractual environment to determine equilibrium allocation. Such an approach is also inherited by the recent development of monetary search models such as Trejos and Wright [42] and Lagos and Wright [15]. I Chapter 4. Competitive Cooperation 46 reckon it is often the case that a large market contains an environment of enforceable contracts as well as non-enforceable ones, therefore it is through a mix of both that most frictional market equilibrium allocations gets built and maintained.

4.2 Preliminaries

The market is populated by the set of S = {1, ...M} sellers and B = {1, ...N} buyers. Both buyers and sellers are homogeneous among their groups, and I denote the buyer-seller ratio, Φ = N/M as market tightness.

Each player could produce one non-storable good by taking a one-shot eﬀort. A good could not be consumed by its producer, but could be consumed by his partner. The quality of a good is a increasing function of the eﬀort level put in by its producer.

The searching process is a sequential game between buyers and sellers, it consists of the decision that has to be made by each buyer and seller about the investment level that they will put into producing a good and searching for a partner:

1. All sellers move simultaneously and non-cooperatively, making an eﬀort

xj to produce a good before entering the market.

2. All buyers observe investment levels of sellers in the market X = (x1, x2...xM ),

and commit to partner with one seller with probability θij. At this

stage, buyer i’s action is a mixed strategy proﬁle: θi = (θi1, θi2...θiM ). M P where θij = 1. j=1

3. The buyer chooses an eﬀort level to bid (yi), based on the observation

of the number of buyers committed to the same seller bi. Chapter 4. Competitive Cooperation 47

4. In a multiple match, the seller chooses the buyer offering the highest effort level to partner. In a pairwise match, the seller cooperates with the offering buyer.

5. All payoffs are realized as effort levels offered by both parties (Vij,Uij).

The consequence consists of the level of effort that the sellers and buyers choose and implement. For a realized seller-buyer pair (s, b), the consequences are represented by the reflection from themselves (s, b) to their effort levels 2 (x, y). Mathematically: Cm : {(s, b) → (x, y): s ∈ S, b ∈ B, (x, y) ∈ R+}. For sellers, since production is done before they enter the market to sell, whether they have a partner or not, the cost incurred is already sunk; therefore, for an off-matched s with an effort level of x, there is no action taken from his partner, which shows: Cv(s): {(s, 0) → (x, 0) : s ∈ S, x ∈ R+} . On the other hand, there is no production requirement for buyers when they enter the market, so an off-match simply means that he is not chosen by the seller that he committed to, in which case he is left with his original situation, and this shows: Cv(b): {(0, b) → (0, 0) : b ∈ B}

For any realized match, payoff of each player is given by a function of the consequence set. This defines the payoff as a map from the consequence (x, y) to their utilities (V,U). For each player, the payoff from the search is the utility he gets from his partner’s effort, deducts the effort level he input. Mathemat- ically, I define the payoff P : {(x, y) → (V,U): V (x, y) = v(y) − x, U(x, y) = u(x)−y} with a uniformed cost function: c0(·) = 1 2, and production functions: 0 0 00 00 0 0 v (0), u (0) > 1 , v (·), u (·) 6 0 and v (∞), u (∞) < 1 which guarantees that the total surplus is finite. This payoff form captures the idea of externality. One’s benefit can only be realized by their partner’s costly effort transferred into his utility. I initialize the origin (0, 0) as the original situation, Mathe- def matically, P : {(0, 0) → (0, 0)} .

2This setting diﬀers from usually assumed by monetary search models, in which both costs and beneﬁts take a non-linear form, but it actually does not lose any generality. See Appendix for mathematical analysis. Chapter 4. Competitive Cooperation 48

4.2.1 On Commitment

Peters (1984) used ”limited mobility” to rationalize a buyer’s behavior to commit on a partnership with a seller. Most current directed search literature has inherited this rationality, and assumes a contract to be fully enforceable. An implicit assumption however, needs to be made to guarantee enforceability. By sending one seller an offer, the buyer not only commits to partner with the particular seller other than anyone else, but also commits to accept the terms and conditions written by the seller, whenever the seller chooses to couple with him. In the situation presented here, this implicit assumption is difficult to accept because seller’s cost is sunk and the information structure embeds a possibility of renegotiation. In particular, instead of putting forward an enforceable contract which reflects the payoff to buyers, the seller now presents the existing results of his effort to attract buyers in the first stage, when the cost involved is already sunk. Once a buyer is attracted by one seller’s project, one can assume that some kind of restraints are imposed on him so that it is hard to switch to another seller from this stage on. This is the process that I refer to as ”commitment”, and it can be made in different ways. In physical terms, we can assume that a constraint is imposed on the buyer from the time that he chooses one seller to deal with3, and this is precisely the situation Peters [28] referred to as ”limited mobility”. An equivalent situation which describes this commitment in legal terms can be an exclusive offer sent from a buyer, stating that the buyer is willing to cooperate with this particular seller over anyone else. After this stage, if it is verified by the court that he did cooperate with someone else, then some punishment that he cannot afford will be imposed. However, the effort level of the buyer at this stage is uncertain, as it relies on buyer’s sequential actions which depend on the observation of a number of competing buyers.

3For example, move to an island of one seller which is far from other sellers. Chapter 4. Competitive Cooperation 49

4.2.2 Local Competition: Bidding Auction

At the beginning of stage three, once a buyer commits to deal with one seller, the ”local” market is then a perfectly competitive market. Given the sellers’ eﬀort level xj, two factors about this local market are crucial to determine equilibrium bid y: (1) The number of buyers who commit to seller’s program and (2) Both parties’ opportunity cost.

Denote bj as the number of buyers committed to cooperate with seller j; clearly, there are three possible scenarios for each seller:

  = 0  bj : = 1   > 1

For bj = 0, the seller is left with a vacancy, and no further payoﬀ will occur. For an individual seller, this marks the end of the game, and the payoﬀ yields: v Vj = 0 − xj = −xj.

When bj = 1, the seller is met with one buyer. Since the seller’s investment happens ex-ante and cost has been sunk, the seller is ”held-up” by the buyer. In a complete information environment, the price that a buyer will be charged is the minimum effort that the seller will accept, which is denoted as yp. Precisely, this minimum effort will make seller indifferent to be matched or off-match, which mathematically shows: v(yp) ≡ 0. Seller’s payoff then yields: p v Vj = 0 − xj = Vj .

As for bj > 1, seller is facing multiple buyers’ offers. The seller is therefore able to hold-up the property and let the buyers compete by offering their efforts. This gives rise to a situation of a standard ascending-bid auction game (see for example, McAfee and McMillan [19]). With complete information and homogeneous buyers, the Bertrand competition between buyers drives the equilibrium offer up to a level which makes buyers indifferent between Chapter 4. Competitive Cooperation 50 getting the deal and any other outside option. Denote this effort level as ym, clearly ym = u(x). Then the seller’s payoff in multiple matches yields: m m m Vj = v(y ) − xj where y = u(x).

4.3 The Model

I attempt to derive symmetric equilibria in a ﬁnite market, and then extend to an inﬁnite market in this section. I use backward induction to derive agents’ optimum action in each stage, and I proceed from the simplest case of N = 2 and M = 2, and then extend to more general cases. This approach follows micro founded directed search literature such as [4], I found this approach tractable since it captures rich thoughts within the game, while still being easy to handle.

4.3.1 The 2 × 2 Case

I start from the buyer’s visiting strategy at stage two by observing sellers’ effort vector in the market x = (x1, x2). Denote Uj as the benefit that the buyer will get from committing to seller j, and we consider buyer i’s visiting strategy, where i, j = 1, 2. If buyer 1 is the only one who visits seller 1, then he is able enjoy the product with paying seller zero. Otherwise, if both buyers visit the same seller 1, then the bidding auction will make the buyers indifferent to whether or not they get the deal. For any given seller’s effort level x , this yields: 1   u(x1), b1 = 1 U1 =  0, b1 = 2

Let Uij denote the (ex-ante) expected utility that buyer i visits seller j, and

θi denotes the probability that buyer i visits seller 1: Chapter 4. Competitive Cooperation 51

U11 = (1 − θ2)u(x1)

Similarly, U12 = θ2u(x2), buyer 2’s payoﬀs are symmetric. We seek mixed strategy Nash equilibrium in this sub-game. For buyer 1, he solves the problem of maximizing expected utility:

maxEU1(θ1 | θ2) = θ1U11 + (1 − θ1)U12 hθ1i

= θ1(1 − θ2)u(x1) + (1 − θ1)θ2u(x2)

Straight implication of rationality is that, buyer 1 will choose seller 1 with probability 1 if U11 > U12, and vice versa (Pure strategy). Buyer 1’s probability choice of visiting seller 1 yields:

  0 if U12 > U11  θ1 1 if U12 < U11   [0, 1] if U12 = U11

The unique symmetric Mixed Strategy Nash Equilibrium yields:

u(x1) ∗ θ1 = θ2 = ≡ θ u(x1) + u(x2)

This demonstrates the buyer’s decision rules. Implication of this result are as follows: First, the unique mixed strategy profile is proportional to the benefit generated by sellers’ effort level. As long as the seller inputs a positive effort level xj > 0, in equilibrium the buyer will always give a positive weight in mixed strategy profile to choose the seller. Second, two pure strategy equilibria exist for this game: (θ1, θ2) = (0, 1) or (1, 0), which are referred to as ”coordinate equilibrium” in which sense both buyers understand each other’s choice and react coordinately; while θ∗ is the equilibrium in which buyers choose their probability of visiting independently. As has been discussed in Chapter 4. Competitive Cooperation 52

[4], it is reasonable to ignore these pure strategy proﬁles as equilibrium, and focus on an symmetric mixed strategy especially in large markets with a lack of coordination.

∗ Now I take the buyers’ mixed strategy equilibrium proﬁle θ1 = θ2 ≡ θ in stage two as given, and consider the sellers’ producing strategy in stage 1. As analyzed before, seller j’s payoﬀ depends on the number of buyers who commit to his program denoted as bj, and this shows:

  −xj, if bj = 0, 1 Vj = . m  v(yj ) − xj, if bj = 2

Conditional on seller 2’s production with eﬀort level x2, seller 1 solves the problem:

∗2 m ∗2 maxEV1(x1 | x2) = θ (v(y1 ) − x1) + (1 − θ )(−x1) (4.1) hx1i 2 u(x1) m = v(y1 ) − x1 (4.2) u(x1) + u(x2)

m Where y1 = u(x1) is deﬁned as before, and seller 2’s problem yields symmetry. For each seller j, the ﬁrst order conditions of sellers yield:

2 m u(xj) 2u(x−j) v(yj ) 0 m 1 + v (yj ) = 0 u(xj) + u(x−j) [u(xj) + u(x−j)] u(xj) u (xj)

The LHS indicates the benefit of an extra effort from the seller. Namely, a marginal increase in the seller’s effort will benefit the seller in two ways. First, it increases the probability that both buyers will choose him, in which case he m will get utility v(y1 ). Second, it also increases the maximum bid of the buyer m yj in such cases. On the other hand, the RHS indicates the marginal cost of such an adjustment. Chapter 4. Competitive Cooperation 53

The uniqueness of the Nash Equilibrium requires the expected payoff function to be quasi-concave, this of course, will require further restrictions to the payoff structure. Although the asymmetric equilibria are potentially interesting, they involve the same coordination problems in a large market as they would in the second stage buyers’ visiting game. Suppose xj1 and xj2 are the equilibrium effort levels of two homogeneous sellers, to finally implement these levels separately, they need to coordinate on the same equilibrium choice as each other, which is usually impractical. Therefore I focus on the symmetric equilibriums of sellers’ actions throughout this paper.

By the above condition, it is now the time to establish the following proposition:

Proposition 4.1. The symmetric Nash equilibrium action in stage one x∗:

• it is well deﬁned on x∗ ∈ (0, x¯1) iﬀ. v0(0)u0(0) > 2.

∗ ∗ 0 ∗ 4u(x∗) • it satisﬁes: v(u(x )) + u(x )v (u(x )) = u0(x∗) .

Proof. The condition v0(0)u0(0) > 2 guarantees the existence of equilibria in the area (0, x¯1). I follow the track that ﬁrst proves that the assigned point of x∗ belongs to the group of symmetric Nash Equilibrium, and then prove its uniqueness. See appendix for details.

4.3.2 The N × M Case

I extend the existing model to the case that the market consists of N buyers and M sellers. Again, I focus on symmetric equilibria where all buyers adopt the same mixed strategy.

Consider the second stage game. Upon observation of sellers’ announcement vector (u(x1), u(x2), ...u(xM )), an arbitrary buyer i visits seller j with possibility θij, according to his expected utility of doing so: Chapter 4. Competitive Cooperation 54

Uij = Ωiju(xj)

where Ωij denotes the probability that he is the only one who visits the seller conditional on the fact that buyer i visited him. Clearly, this probability purely relies on other buyers’ choice of probability, which is shown:

N Y Ωij = (1 − θtj) t=1,−i

For each individual buyer i, he opts his mixed strategy to maximize his expected payoﬀ, the problem yields:

( M ) X max Ui = θijΩiju(xj) hθij i j=1 M X s.t. θij = 1 j=1

Lemma 4.2. In symmetric mixed strategy equilibrium, conditional on sellers’ posted utility level u = (u(x1), u(x2), ...u(xM )), buyers will choose seller j with the same probability, which is: θlj = θkj = θj for any l 6= k. Furthermore, for any xp = xq, the equilibrium condition yields: θp = θq.

Proof. See appendix.

Straight implication of this lemma comes in a symmetric equilibrium, sellers post the same level of x :(x, x, ...x), buyers will visit sellers with the same 1 probability: θij = M where i ∈ (1, 2...N), j ∈ (1, 2...M).

Now consider the seller’s ﬁrst stage equilibrium conditional on buyers’ decision rule. Seller j’s problem given other sellers’ choice ¯x−j yields: Chapter 4. Competitive Cooperation 55

   V (xj | ¯x−j) =  max E N N−1 m  −xj + 1 − (1 − θj) − Nθj(1 − θj) v(yj )    Ωju(xj) = Ω−ju(x−j) = λ; M s.t : P  and θj = 1 .  j=1

The solution to the seller’s problem is shown by the following Proposition:

Proposition 4.3. In a market populated by N × M, the symmetric Nash equilibrium in stage one x∗:

• it is well deﬁned on x∗ ∈ (0, x¯1) iﬀ.

1 v0(0)u0(0) > h N M−1 N N M−1 N−1i 1 + M − 1 M − M M

• it satisﬁes:

1 = u0(x∗) " # M − 1N N M − 1N−1 1 − − v0(u(x∗)) M M M N M − 1N v(u(x∗)) + M M u(x∗)

Proof. See appendix.

The implication of this proposition come as follows. First, it addresses the condition of existence of the Nash equilibrium. Namely, since the possibility that the seller’s effort level will be held up always exists, a positive level of effort relies on a market structure that yields large enough probability that sellers will get the surplus. Second, it addresses the area of the potential equilibrium as (0, x¯1), so that I can further analyze the social properties of this effort level and compare this with the bargain solution. Chapter 4. Competitive Cooperation 56

4.3.3 Large Market

Conditions of finite markets can be extended to large market in their limit form. As before, I seek for symmetric equilibria with all sellers posting the same x, and buyers use the same mixed strategy θi(x). The mixed strategy is directed by sellers’ vector x, on and off-equilibrium path. The mixed strategy generates a binomial distribution of buyers over sellers, and hence, an endogenous matching function. The probability that a seller will M n N−n be selected by n buyers given, is Pr{n|x} = n (θi(x)) (1 − θi(x)) . In any symmetric equilibrium, since all sellers will also post the same x, then 1 θi(x) = M , ∀i ∈ M, when M,N goes to infinity with a constant ratio Φ, the Φne−Φ matching rate shows: limN,M→∞ Pr{n|x} = n! . Sellers face a Poisson ar- PN Φne−Φ −Φ rival of buyers. Since n=0 Pr{n|x} = n! = 1, hence Pr{n = 0|x} = e , and Pr{n = 1|x} = Φe−Φ. Since it is well known that directed search taken to the limit generates the same outcome as competitive search, I simplify the analysis and focus on the competitive search construct, but with a Poisson matching technology.

Consider one deviant seller postingx ˆ, while all other sellers post x. The program for a deviating seller is

n o V = max (1 − φeˆ −φˆ − e−φˆ)v(ym) − xˆ (4.3) ˆ

(1 − φeˆ −φˆ − e−φˆ) s.t.Uˆ = e−φˆ[u(ˆx) − yp] + [u(ˆx) − ym] ≥ U¯ (4.4) φˆ ˆ ˆ where φ = φ(ˆx, x−1) 6= Φ is the expected ”queue length” generated by the de- viationx ˆ 6= x. The utility level U¯ is the maximum expected utility a buyer can obtain from any other sellers in the market. In the competitive search literature, this is referred to as the market utility property. A deviating seller takes this maximum utility as given, and the deviation creates an adjustment in expected queue length, which operates much like a standard demand function following a price reduction. Chapter 4. Competitive Cooperation 57

I extend the model to its limit form and derive the following proposition for an inﬁnite market:

Proposition 4.4. In a market where N,M → ∞, the symmetric Nash equilibrium action in stage one x∗:

∗ 1 0 0 1 • it is well deﬁned on x ∈ (0, x¯ ) iﬀ. v (0)u (0) > 1−e−Φ

• it satisﬁes:

Φv(u(x∗)) eΦ + (eΦ − Φ − 1)v0(u(x∗)) = . u(x∗) u0(x∗)

Proof. See appendix for details.

This proposition addresses the circumstance where a market could exist in presence of the hold-up problem. Since in many markets there exists such a hold-up problem, sellers are discouraged to invest in advance unless the market tightness is sufficiently high, so that they will have enough probability to get surplus. For a particular payoff structure u(·) and v(·), a ”benchmark” market tightness exists,so that under this market tightness sellers will not h i ˆ 1 ˆ start their investments. Explicitly, Φ = − ln 1 − v0(0)u0(0) where Φ denotes the least market tightness that enables sellers to have a positive expected value to enter the market. In the other way around, where a market has a large number of buyers and sellers and the market tightness is fixed, the initial marginal product should be large enough to start such a market.

4.4 Characterizing Equilibrium

This section attempts to characterize the equilibrium. On the ”intensive margin”, I characterize equilibrium action pairs (x∗, y∗) in aspects of comparative statics and eﬃcient properties. On the other hand, we also characterize the properties on ”extensive margins” such as equilibrium matches and free entry. Chapter 4. Competitive Cooperation 58

4.4.1 Comparative Statics

I ﬁrst evaluate if the equilibrium x∗ is increasing or decreasing in Φ, this is illustrated by the following proposition.

Proposition 4.5. The equilibrium eﬀort x∗ is increasing in market tightness, and it converges to x¯1 when market tightness goes to inﬁnity:

dx∗ > 0 and x∗(Φ) | =x ¯1 dΦ Φ→∞

Proof. See appendix.

This seems a bit counter-intuitive at first glance: since sellers will pre-commit to their effort, which is costly, only the buyer will benefit; and when market tightness becomes larger, the seller has more choice and it is the buyers who become more competitive among each other. Why does the seller have the incentive to increase their effort?

The trick lies in the payoff dispersion structure. In a multilateral match, buyers will do his best to win the partnership, and in the end, the effort that ∗ they attempted will cost the same as the seller’s offer (u(ˆx) = c(ym)). The surplus of the cooperation is fully extracted by the seller, the more they offer in the first round, the more surplus they are supposed to get in a multiple match. However, they will always under-cooperate unless the market tightness goes to infinity, in which case the probability they get in a multiple match goes to 1. This is out of fear of the possibility of showing up in a pairwise match, in which case they will get nothing in return for already sunk costly effort. As a balance, a seller’s optimum choice is always to offer a cooperation level that is lower than the optimum level in a multiple match. When market tightness gets larger, the probability of a pairwise match is lower and the probability of a multiple match is higher, so the seller’s optimum offer increases. It finally approaches the optimum level that they will offer in a multiple match as the Chapter 4. Competitive Cooperation 59 market tightness goes to infinity, and the probability of a multiple match goes to 1.

To conclude this section, I state that with higher monopoly power, sellers are willing to commit to a higher level of cooperation, and this cooperation level converges to an efficient level when market tightness goes to infinity. The monopoly market is the most likely efficient market structure in this frictional environment. Such a model explains the observations of under cooperation in a market where initial investment is a sunk cost.

4.4.2 Matching

The matching probability of each buyer after committing to one seller’s program highly relies on the buyer’s subsequent behavior, and is distinctly different from what can be derived in standard directed search models, commonly known as ”urn-ball matching” technology. It is worth noting that, instead of considering matching probability only as in standard directed search models, I further explore the ”intensiveness” of each match, since it is now necessary to distinguish different levels of effort in a pairwise match and a multiple match.

The expected number of successful buyer-seller matches, which is known as ”matching function” yields:

1 Γ∗(N,M) = M 1 − (1 − )N M

Among them, the number of multiple matches, which yields an auction solution in equilibrium ym = u(x∗), and sellers get positive payoﬀ yields: Γm(N,M) = h 1 N N 1 N−1i M 1 − 1 − M − M 1 − M

The number of pairwise matches, which yields the action pair (x∗, 0) in equi- p h N 1 N−1i 1 N−1 librium is shown: Γ (N,M) = M M 1 − M = N 1 − M Chapter 4. Competitive Cooperation 60

The probability of a successful match for a particular individual, which is commonly referred to as ”arrival rate” in directed search models, is Ab = Γ∗(N,M) M 1 N N = N 1 − (1 − M )

Γ∗(N,M) 1 N and As = M = 1 − (1 − M ) for buyers and sellers, respectively.

These arrival rates are similar results to the standard directed search models, with our particular game structure, I are also interested in the amount that each agent will improve their welfare. For an individual buyer, the ”beneficial rate”, which is defined as the probability that he will be better off in a pairwise match, and therefore get surplus from entering the market, yields:

Γp(N,M) 1 N−1 B = = 1 − b N M

For individual seller, the ”beneficial rate” is defined as the probability that he will get multiple offers, yields:

" # Γm(N,M) 1 N N 1 N−1 B = = 1 − 1 − − 1 − s M M M M

Fixing Φ = N/M and N,M → ∞, the matching function in a large market yields: τ ∗(Φ) = M 1 − e−Φ and :

1 − e−Φ A = ; A = 1 − e−Φ b Φ s −Φ −Φ −Φ Bb = 1 − e ; Bs = 1 − Φe − e given any market tightness Φ, the arrival rate is uniquely derived from this form. In terms of both arrival rate and beneﬁcial rate, the properties of ”ap- proximately constant returns to scale” in the previously mentioned standard Chapter 4. Competitive Cooperation 61 directed search models are maintained here.

4.4.3 Under-investment: Hold-up Problems in Directed Search

Results of my model support the commonly held view that a hold-up problem leads to a situation of under-investment. Namely, being uncertain of a sufficient share of the return in the project, the investor tends to input an insufficient (and hence inefficient) level of investment. In particular, a seller entering such a market with an investment level x will consider a combination which consists of two cases: getting a positive payoff V m = v(u(x)) − x with p v a probability of Bs, or getting a negative payoff V = V = −x with a probability of 1 − Bs, which is always a positive measure. Compared to the P AR set I defined in the previous chapter, seller’s investment level is driven down by this positive probability of losing out.

To illustrate the effect of such hold-ups in our model, I consider the following situation in large market to compare. Suppose sellers have the access to an outside option which gives them a zero payoff in both vacancy and pairwise matches, V p = V v = 0. Of course, this will affect buyers’ payoff in a pairwise match as well, since now instead of bidding yp = 0, buyers in pairwise match −1 have to bid to a level ofy ¯p = v (x). In this case, sellers solve the problem:

n o V = max (1 − φeˆ −φˆ − e−φˆ)[v(ym) − xˆ] (4.5) ˆ

ˆ −φˆ ¯ s.t.U = e [u(ˆx) − y¯p] ≥ U (4.6)

−1 Where ym = u(x) andy ¯p = v (x).

Solving this model with the same technology, the equilibrium condition yields:

0 ∗ 1 ∗ ∗ Φ [u (¯x ) − 0 ∗ ][v(y ) − x¯ ] (e − Φ − 1) v (¯yp) m = ∗ ∗ 0 ∗ 0 ∗ (4.7) Φ [u(¯x ) − y¯p] (1 − v (ym)u (¯x )) Chapter 4. Competitive Cooperation 62

∗ ∗ ∗ −1 ∗ with ym = u(¯x ) andy ¯p = v (¯x ).

The existence and comparative statics of this model is described as the following proposition:

Proposition 4.6. For any market tightness Φ ∈ (0, ∞) :

• x¯∗ which satisﬁes condition 4.7 is well deﬁned on (¯x1, x¯0);

dx¯∗ ∗ 1 • dΦ < 0 and x¯ (Φ) |Φ→∞=x ¯

Proof. First, checking the limits of RHS which derive:

lim RHS = ∞ 1 x¯→xˇ+

lim RHS = 0+ 0 x¯→x˜−

While LHS is a positive ﬁnite value strictly increasing in Φ, and RHS is strictly decreasing inx ˆ. This guarantees that there exists a uniquex ¯∗ ∈ (¯x1, x¯0), and it decreases with Φ, see appendix for details.

The following intuitions come from comparing this equilibrium with our model. First, equilibrium levels of efforts in both multiple matches (x∗, u(x∗)) and pairwise matches (x∗, 0) belong to an area of under- investment UI. Second, the equilibrium investment derived from our model is visibly lower than any Pareto efficient level of effort (x∗ < x¯1), mostly because the positive measure of negative payoff (Hold-ups). As I illustrated in this section, eliminating the possibility of a negative payoff will result in a higher level of pre-investment, which is conjecturally more efficient. (¯x∗ > x¯1). Finally, the under-investment in the first round from sellers actually gives rise to buyers’ under-investment in the second round in a multiple match, comparing with the action pair of multiple-match I derived in this chapter (¯x∗, u(¯x∗)) which arrives in OI, the buyer’s auction effort is lower because of insufficient pre-investment from the Chapter 4. Competitive Cooperation 63 seller in the first round. To sum up, the hold-up problem which gives rise to a potentially negative payoff substantially discourages sellers from a higher level of pre-investment, and this under-investment level discourages buyers’ effort level in the bidding auction, and as a result, actions taken in both pairwise matches and multiple matches appear in UI.

4.4.4 Free Entry

My model also allows endogenous market tightness via free entry. I focus on the case of an inﬁnite market in this section. Suppose the cost of entry is ¯ ¯ endogenously given by k > 0. Clearly, if k = 0, then the free entry market tightness Φ(¯ k¯ = 0) is the least market tightness a ﬁrm will choose to enter the market:

1 Φ(¯ k¯ = 0) = Φˆ = − ln 1 − v0(0)u0(0)

Now suppose that every ﬁrm has to pay a positive entry cost k¯ to enter the market. I consider equilibrium entry which satisﬁes the condition:

¯ ∗ ¯ ¯ ∗ −Φ¯ ∗ −Φ¯ ∗ ∗ ∗ ¯ Φ (k) : (1 − Φ e − e )v(ym) − x = k (4.8)

∗ ∗ ¯ ∗ Both ym and x are functions of Φ , the RHS is a single value function of Φ¯ ∗, this equation gives equilibrium entry Φ¯ ∗. Free entry properties can be described by the following proposition:

Proposition 4.7. For any entry cost k¯ ∈ [0, v(u(¯x1)) − x¯1), the market tightness Φ¯ ∗(k¯) which satisﬁes 4.8:

• is well deﬁned.

dΦ¯ ∗ ¯ ∗ ¯ • dk¯ > 0 and Φ (k) |k¯→x¯1 → ∞ Chapter 4. Competitive Cooperation 64

Proof. See appendix for details.

The free entry condition describes the benchmark market tightness should the seller choose to enter the market, and the entry cost of a particular market structure. The seller should make the decision whether or not to enter the market according to the entry cost incurred and the market tightness. In reality, a large proportion of the entry cost could be regulated by governments, such costs includes investment tax, licensing fee, etc. Given the free entry condition, governments could adjust the industrial structure by regulating the relative entry costs throughout diﬀerent markets, therefore optimizing the eﬃciency of the economy.

4.5 Application

Goods Exchange Market with Monetary Transfer

My model has a standard goods market application with costly production ex-ante. Consider the basic environment of Burdett et al. [4]: Each seller wants to sell a unit of product which costs 0 to produce, but generates 1 unit of utility for the buyer. If the buyer pays the seller the price of p, then he will receive the utility U = 1 − p, and the seller gets a surplus of V = p. It is clear that this set up does not fit our framework perfectly because of uniform utility. However, the following modification can be made to fit into my framework: extend the value functions to the form of u(x) = 1 − e−αx, and v(y) = y. Therefore, the buyer’s value function: U = 1 − e−αx − y, and seller’s value function: V = y − x. where y is the equivalent of the price paid by the buyer (a cost), which provides benefit to the seller (revenue).

Since eﬀorts of buyers take a form of linear function, this model have a linear Pareto frontier: Chapter 4. Competitive Cooperation 65

P AR : u0(¯x)v0(¯y) = 1 and u(¯x) − y,¯ v(¯y) − x¯ > 0 ln α ln α 1 ⇒ x¯ = , y ∈ , 1 − α α α

The P AR set is represented by a vertical linear section on the action pair ln α 1 ln α space: P AR : (¯x, y¯) :x ¯ = α , 1 − α > y¯ > α

Using my model, the equilibrium condition is:

eΦ − 1 eαx∗ = . eΦ α initial productivity level:

1 1 = < ∞. v0(0)u0(0) α

Not any market tightness will carry a positive pre-investment in our model.

Notice if α 6 1: 1 Φˆ = − ln(1 − ) → ∞, α then no market tightness will give the seller enough of an incentive to produce and enter the market. The practical meaning of this situation is that, compared to the cost incurred during production, the utility generated from this kind of good is not sufficient to start a market, even though there are many people who need these kinds of goods. There is one observation which particularly fits this situation in reality; when comparing to some highly decentralized goods such as sports cars and hi-fi systems, the variety along with the quality provided in daily necessities such as toothpaste and tissue are visibly low, even though there is a far higher demand for them. In such cases, sellers are not willing to input high levels of investment into innovation and Chapter 4. Competitive Cooperation 66 advertisements because consumers do not value such commodities. Although they have a high probability to be sold, the revenue is relatively slim.

Consider α > 1, and free entry condition is satisﬁed:

1 Φ > Φˆ = − ln(1 − ) α there is unique equilibrium x∗ which the seller will produce:

" # 1 α eΦ − 1 x∗ = ln α eΦ

dx∗ Clearly, dΦ > 0 as I veriﬁed before. Also, when market tightness goes to inﬁnity:

ln α lim x∗(Φ) = Φ→∞ α then I get:

ln α x∗ | = Φ→∞ α

ˆ 1 The seller will start to produce if market tightness Φ > Φ = − ln(1 − α ), and if market tightness goes to inﬁnity, the maximum production of the seller is ln α α .

Action pairs of equilibrium: In multiple matches, the action pair:

" # " # 1 α eΦ − 1 1 α eΦ − 1 (x∗, ym):( ln , ln ) α eΦ α eΦ

−Φ −Φ and quantity of this multiple match pairs: MBs = 1 − Φe − e M

While pairwise action pair: Chapter 4. Competitive Cooperation 67

" # 1 α eΦ − 1 (x∗, yp):( ln , 0) α eΦ

−Φ −Φ and quantity: MBp = Φe M, and there are also Me sellers left with vacancy.

4.6 Concluding Remarks

This paper provides a framework to analyze hold-ups caused by externality and sequential investment in a directed search environment. The payoﬀ structure is a continuous Prisoner’s Dilemma game and captures externality in cooperative behaviors. It derived a generalized Nash bargaining solution if the contract is fully enforceable.

I argue that when payoff embeds externality and investments are made sequentially, which is often the case in daily practice of trading, it is difficult to maintain a fully enforceable contractual environment. In the context of the literature on directed search models, it requires additional assumptions apart from the commonly held ones such as ”limited mobility” and ”capacity constraint”. Therefore, in the presence of externality and corresponding hold-ups situations, how a market is organized in a directed search fashion, along with equilibria form and properties under such environments, could be an important topic. In particular, observing that the deviation of payoff structure caused by sequential movement of investment could potentially influence matching probability, the corresponding properties of equilibrium becomes interesting.

In the presence of a positive probability of hold-up situations which yield negative payoffs, sellers are discouraged to entering the market until market tightness reaches a level that bears a positive economic profit. This benchmark market tightness corresponds to the payoff structure, namely, the initial marginal productivity of both parts. I first characterized the type of link with the benchmark market tightness and the initial marginal productivity, and Chapter 4. Competitive Cooperation 68 proved the uniqueness of symmetric equilibrium for any market that has a higher buyer-seller ratio than the benchmark market tightness.

Though higher levels of pre-investment could potentially lead to a more severe hold-up situation when there is a higher market tightness, giving seller a better bargaining situation; sellers want to invest more if the market tightness is larger. This property looks a bit counter-intuitive, especially since it conﬂicts with existing bargaining models. I explain this phenomenon with that when market tightness is larger, the probability of getting surplus, along with the surplus itself, will be larger; thus, sellers are encouraged to invest more. However, from a more general view, the pre-investment level was lower than any of the pre-mentioned bargaining solution contracts, this supports the commonly held view that incomplete contract will lead to under-investment. Free entry properties show that a higher capital investment barrier will require higher market tightness in equilibrium, which enables governments to regulate industrial structure by adjusting pre-investment barriers among industries.

My model applies to a considerable proportion of goods market exchanges with monetary transfers, as in the application section, I provide an extension of [4] and illustrate some numerical properties. It is worth noting though, that imperfect contractual environments are not only caused by externality and the possibility of renegotiation, but could be cause by many other factors. For example, partially incomplete information breaks assumptions in Rogerson [36], and could potentially lead to a situation similar to the one described by Myerson–Satterthwaite theorem in frictional markets. Further, my model applies to the case where goods are non-storable; it is plausible to believe, however, that if the products of the seller are assumed to be storable, it will improve the equilibrium eﬀort since the seller could now carry the products to the next period. Whether or not this eﬀort level converges to a bargaining solution when a product is storable, remains an interesting topic. After all, I suggest that situations of imperfect contractual environments are important Chapter 4. Competitive Cooperation 69 to frictional markets, and the one we analyzed here caused by production externality and sequential investment, could be interesting. Appendix A

Mathematical Proofs

Theorem 2.3:

Proof. First, the following lemma can be established:

∗ Lemma. For any contract posted in equilibrium Ci = {Xil}l∈Ω, the state- contingent action s.t.: P (Xil) ∈ PO for any l ∈ Ω.

∗ Proof. Suppose in equilibrium, male i post Ci = Xi , in which there exist one 0 0 0 ¯ l ∈ Ω, s.t. Xil :(Vil,Uil) ∈ P/PO, then there exist Xil :(Vil,Uil) ∈ V (U) s.t. 0 0 Vil > Vil, Uil > Uil. This means, male could maintain the female’s payoff in 0∗ 0 any state, and obtain better payoff by posting Ci = Xil, Xi(−l) l∈Ω. By this deviation, since female’s payoff is maintained in any states, it wouldn’t change female’s equilibrium visiting strategy, therefore it is a profitable deviation from ∗ 0∗ ∗ Ci to Ci . Ci is not equilibrium contract. QED for the Lemma.

Now to prove the theorem, I consider the general circumstance that l could be any match-contingent or non-match-contingent state. The diﬃculty is to formulate the probability of l which is endogenously depends on female’s visiting strategy. Without generality, I denote probability of every state as a function of equilibrium visiting strategy to capture the (potentially) match contingent ∗ probability: ql(θj ).

70 Appendix A Mathematical Proofs 71

For male i, suppose that given other males’ choice C−i , the optimum strategy of i is to post a list {Xil}l∈Ω where Xil1 6= Xil2 , and this yields the females visit this male with probability θ∗ = θ∗ in the sub-game mixed strategy i ji j∈F ∗ P ∗ P ∗ equilibrium. Male’s payoﬀ then yields: ϕi(θi ) l ql(θi )Vl , where l ql(θi ) = ∗ 1, and ϕi(θi ) is the matching probability that male is visited by at least one of female. For any female j, visiting this male receives expected utility ∗ P ∗ ∗ ϕj(θi ) l ql(θi )Ul, where ϕj(θi ) denotes the probability that j is chosen from i’s visitors, conditional on j visits i.

¯ 00 ˜ ¯ ˜ P ∗ ˜ Given that V (U) < 0, there exist a X ∈ V (U) s.t. V > l ql(θi )Vl, and U = P ∗ l ql(θi )Ul, thus the male could get higher expected payoﬀ while maintaining ˜ female’s payoﬀ (and thus visiting strategy) by switching from {Xl}l∈Ω to X. ˜ {Xl}l∈Ω is dominated by X. QED.

Theorem 2.7:

Proof. According to Theorem 2.3, males all post contract C = X :(V,U) ∈ V¯ (U) in equilibrium. Therefore the contract could be uniquely represented by the utility level U they promise to provide for females. Consider the second stage game. Upon observations of males’ announcement vector U =

(U1,U2...UM ), a female j visit male i with possibility θji according to his expected utility:

Uji = ΦjiUi where Φji denotes for the probability she ﬁnally deals with the male conditional on she visited him. Clearly, this probability purely relies on other females’ choice of probability, which shows:

Φji =

F  F  F  F  F  X 1 X  X X Y  θ θ ... θ (1 − θ ) n  t2i t3i  tni  t1i   n=1 t2=1,−j t3=1,−(j,t1,t2) tn=1,−(j,t1,t2...tn−1) t1=1,−(j,t2...tn)  Appendix A Mathematical Proofs 72

For each individual female j, she manage her mixed strategy to maximize her expected payoﬀ, the problem yields:

( M ) X max Uj = θjiΦjiUi hθj i i=1 M X s.t. θji = 1 i=1

Let L denotes the Lagrangian:

M X L = Uj + λ(1 − θji) i=1 and F.O.C yields:

∂L = 0 : ΦjiUi = λ for any i; (A.1) ∂θji M ∂L X = 0 : θ = 1; ∂λ ji i=1

Conditions (A.1) gives the decision rules of the directed search equilibrium. Now I establish that any non-coordination equilibrium is symmetric equilibrium if exist by the following lemma.

Lemma In the sub-game after males post their contracts U = (U1,U2...UM ), females choose male i with same probability, which shown: θj1i = θj2i = θi for any j1 6= j2.

Proof. Omitting potential coordination equilibriums that have pure visiting strategies in equilibrium, I now focus on mixed strategy equilibrium, and seek for inner solution of Lagrangian. As shown, female j select visiting strategies by condition:

Φj1iUi = λ for any i ∈ (1, 2, ...M)

where Φj1i is a strictly decreasing function of any θ(−j1)i including θj2i. Appendix A Mathematical Proofs 73

Suppose θj1i > θj2i, which directly implies:

λj1 = Φj1iUi < Φj2iUi = λj2

M M 0 P P 0 Notice that θj1p = θj2p = 1, there exists at least one i 6= i, s.t. θj1i < p=1 p=1 0 θj2i , this implies:

0 0 0 0 λj1 = Φj1i Ui > Φj2i Ui = λj2

This conﬂicts with (A.1). Therefore assume a mixed visiting strategy equilibrium exists, then θj1i = θj2i = θi. QED of Lemma.

Now I re-write female’s choice rule as:

ΦiUi = λ F (1 − (1 − θi) ) where Φi = F θi

Clearly, for any Up = Uq, this yields: θp = θq.If in equilibrium, males post the same level of U : U = (U, U, ...U), then females will visit males with same 1 probability: θji = M where i ∈ M, j ∈ F.

Now consider the male’s ﬁrst stage game conditional on females’ decision rule.

Male i’s problem given other males’ choice U¯ −i yields:

F ¯ max E V (Ui | U−i) = (1 − (1 − θi) )V (Ui) M X s.t ΦiUi = Φ−iU−i = λ, and θi = 1 i=1

Suppose every males except for i now post the same utility U−i = (U, U.., U) initially, and male i contemplates deviating to Ui, therefore females visit seller i with probability θi. Then probability that she visits each of non-deviants is ¯ given by: θ = (1 − θi)/(M − 1), therefore, a female who visits the deviant will Appendix A Mathematical Proofs 74

ﬁnally partner with probability:

F 1 − (1 − θi) Φi = F θi and a female who visits a non-deviant is served with probability:

F 1 − (1 − [(1 − θi)/(M − 1)]) Φi = F [(1 − θi)/(M − 1)]

Substitute into females’ decision rule, these yields:

F Ui (M − 1)θi{1 − (1 − [(1 − θi)/(M − 1)]) } = F ≡ Ψ(θi) U (1 − θi) [1 − (1 − θi) ]

Observe that, Ψ(θi) is a strictly increasing function with θi, and also, using L’Hospital:

1 − [(M − 2)/(M − 1)]F lim Ψ(θi) = 6 1 θj →0 F/(M − 1)

lim Ψ(θi) = F θj →1

Ui Therefore, there is a unique reaction of θi(Ui,U) ∈ (0, 1) iﬀ: Ψ(1) > U¯ > Ψ(0).

Ui Ui If U¯ < Ψ(0), no females visit the deviant, and on the other hand, if U¯ > Ψ(1), everyone visit him though the chance of being picked is slim. This shows:

 Ui  0 if U < Ψ(0)  θ Ui i 1 if U > Ψ(1)   Ui  (0, 1) if Ψ(1) > U > Ψ(0)

The seller has no incentive to deviate to Case I, since it yields:

" # 1 F V (U | U) = 0 < 1 − 1 − V¯ (U) i M Appendix A Mathematical Proofs 75

For Case II, it yields:

1 V (U | U) < V¯ (FU) < V¯ (U) i F

1 h 1 F i 1 Assume F < 1 − 1 − M , I ruled out Case II as well. Now I seek equi- ¯ librium oﬀer of deviating male Ui, where his problem yields:

F ¯ max V (Ui | U) = (1 − (1 − θi) )V (Ui) ¯ hUii U s.t. : i = Ψ(θ ) U i

F.o.c:

F ¯ 0 F −1 ¯ ∂θi −(1 − (1 − θi) )V (Ui) = F (1 − θi) V (Ui) ∂Ui F −1 ∂θi 0 F (1 − θi) V¯ (U ) ∂Ui = − i F ¯ (1 − (1 − θi) ) V (Ui)

Observe that LHS is strictly decreasing and RHS is strictly increasing w.r.t. ¯ ¯ ¯ ¯ ∗ ¯ Ui, this guarantees the unique equilibrium being Ui = U = U and θi = θ = ∗ 1 θ = M , which implies: ( ) M 2 M − 1F −1 F − 1 Φ0(θ ) = Φ0(θ ) = 1 + − 1 , i −i F M M substitute in the condition:

1 V¯ 0(U ∗) = − . M M−1 −F M M ∗ ¯ ∗ F ( M ) − F − M−1 U V (U )

This completes the proof.

Proposition 3.1:

1 1 This condition holds for large market. In ﬁnite market, it requires M < 1 , which F −1 F 1−( F ) implies F,M are close in a loose sense. For simplicity, I believe it is reasonable to impose this assumption. Appendix A Mathematical Proofs 76

Proof. Suppose every seller now provides the same quality and charges the same price c1 = c2... = cj−1 = cj+1... = cM = c, p1 = p2... = pj−1 = pj+1... = pM = p initially, and seller j contemplates deviating to Uj, and the buyers visit seller j with probability θj. Then, the probability that he will visit each ¯ of the non-deviants is given by: θ = (1 − θj)/(M − 1), and the buyer who visits the deviant is served with probability:

N 1 − (1 − θj) Ωj = Nθj and a buyer who visits a non-deviant is served with probability:

N 1 − (1 − [(1 − θj)/(M − 1)]) Ω−j = N [(1 − θj)/(M − 1)]

Substitute into buyers’ decision rule, these yields:

N u(cj) − pj (M − 1)θj{1 − (1 − [(1 − θj)/(M − 1)]) } = N ≡ Ψ(θj) u(c) − p (1 − θj) [1 − (1 − θj) ]

By following the analysis from previous chapters, I sought for the symmetric equilibrium action pair of (c∗, p∗), seller’s problem yields:

N max V (cj, pj | c, p) = −cj + (1 − (1 − θj) )pj hcj ,pj i u(c ) − p s.t. : j j = Ψ(θ ) u(c) − p j

F.O.C.s:

∂Vj = 0 : ∂cj 0 N−1 Ω(θj )u (cj ) −pjN(1 − θj) 0 1 0 = 1 [u(cj )−pj ]Ω (θj )+ M−1 [u(c)−p]Ω (θ−j ) ∂V1 = 0 : ∂p1 N−1 N pj Ω(θj )N(1−θj ) (1 − (1 − θj) ) = − 0 1 0 [u(cj )−pj ]Ω (θj )+ M−1 [u(c)−p]Ω (θ−j ) Appendix A Mathematical Proofs 77

Like before, I focus on the symmetric Nash equilibrium to assume cj = c = ∗ ∗ ¯ ∗ 1 c , pj = p = p and θj = θ = θ = M , which implies: ( ) M 2 M − 1N−1 N − 1 Ω0(θ ) = Ω0(θ ) = 1 + − 1 , j −j N M M substitute in the conditions:

1 u0(c∗) = M−1 N 1 − M and: ( ) 1 p∗ = u(c∗) 1 − M M−1 −N M 1 N ( M ) − N − M−1

This completes the proof.

Proof of Lemma 4.2

Proof. Let L denote the Lagrangian:

M ! M X X L = θijΩiju(xj) + λ(1 − θij) j=1 j=1 and the F.O.C. yields:

∂L = 0 : u(xj)Ωij = λ ∂θj M ∂L X = 0 : θ = 1 ∂λ j j=1

Focusing mixed strategy equilibrium, I seek for an inner solution of Lagrangian. (Corner solutions have been ruled out from last chapter). As shown, seller l selects probability of visiting stores by condition:

u(xj)Ωlj = λ for any j ∈ (1, 2, ...M) Appendix A Mathematical Proofs 78

where Ωij is a strictly decreasing function of any θ(−l)j including θkj.

Suppose θlj > θkj, which directly implies:

λl = u(xj)Ωlj < u(xj)Ωkj = λk

M M P P Notice that θkp = θlp = 1, there exists at least one t 6= j, s.t. θlt < θkt, p=1 p=1 which implies:

λl = u(xt)Ωlt > u(xt)Ωkt = λk

This conﬂicts with the Lagrangian. Therefore, in a mixed strategy equilibrium,

I have: θlj = θkj = θj.

Now I are able to re-write the buyer’s choice rule as:

Ωju(xj) = λ

N−1 where Ωj = (1 − θj)

Clearly, for any u(xp) = u(xq), this yields: θp = θq.

Proof of Proposition 4.3.

Proof. Suppose every seller now posts the same utility, x−j = (x, x.., x), initially, and seller j contemplates deviating to xj, and buyers visit seller j with probability θj. Then, the probability that he visits each of the non- ¯ deviants is given by: θ = (1 − θj)/(M − 1), therefore, the probability that a buyer who visit the deviant seller gets a surplus yields:

N−1 Ωj = (1 − θj) and a buyer who visits a non-deviant is served with a probability:

N−1 Ω−j = [1 − (1 − θj)/(M − 1)] Appendix A Mathematical Proofs 79

Substitute into buyers’ decision rule, these yields:

N−1 u(xj) 1 − (1 − θj)/(M − 1) = ≡ Ψ(θj) u(x) (1 − θj)

It is clear that Ψ(θj) is a strictly increasing function with θj. Its limits on each end show:

M − 2N−1 lim Ψ(θj) = 6 1 θj →0 M − 1

lim Ψ(θj) → ∞ θj →1

It is clear that there always exists a unique reaction of θj(x, xj) ∈ (0, 1)

u(xj ) u(xj ) iﬀ: u(x) > Ψ(0). If u(x) < Ψ(0), then no buyers visit the deviant. Since lim Ψ(θj) → ∞, there is no chance that everyone will visit the same seller θj →1 since the seller could never aﬀord such an incident. This shown:  u(xj )  0 if u(x) < Ψ(0) θj u(xj )  [0, 1] if u(x) > Ψ(0)

Since Case I yields zero proﬁt, and that the seller will quit the market, I

u(xj ) assume that u(x) > Ψ(0) for now.

I seek for equilibrium production of deviating seller xj. Seller’s problem yields:

N N−1 m max V (xj | x−j) = 1 − (1 − θj) − Nθj(1 − θj) v(yj ) − xj hxj i u(x ) s.t. : j = Ψ(θ ) u(x) j

F.O.C:

N N−1 0 m 0 N−2 ∂θj m 1 − (1 − θj) − Nθj(1 − θj) v (yj )u (xj)+Nθj(N−1)(1−θj) v(yj ) = 1 ∂xj Appendix A Mathematical Proofs 80

0 ∂θj u (xj ) Where = 0 .and substitute: ∂xj u(x)Ψ (θj )

N−2 m N N−1 0 m Nθj(N − 1)(1 − θj) v(yj ) 1 1 − (1 − θj) − Nθj(1 − θj) v (yj )+ 0 = 0 u(x)Ψ (θj) u (xj)

∗ As before, I focus on symmetric equilibrium that xj = x = x , which yields in ¯ ∗ 1 equilibrium: θj = θ = θ = M , then it yields:

" N N−1# N m M − 1 N M − 1 0 m N M − 1 v(yj ) 1 1 − − v (yj )+ = 0 M M M M M u(x) u (xj)

Given particular number for N and M, the RHS is strictly increasing while the LHS strictly decreasing. So I need to check:

LHS |x∗→0> RHS |x∗→0

Using L’hospital:

" # N M − 1N N M − 1N−1 LHS | = 1 + − 1 − v0(0) x→0 M M M M 1 > = RHS | ∗ u0(0) x →0

Which derives:

1 v0(0)u0(0) > h N M−1 N N M−1 N−1i 1 + M − 1 M − M M

This completes the proof.

Proof of Proposition 4.4.

Proof. Diﬀerentiating the objective function and setting it equal to zero yields

dyˆm e−φˆφˆ0φvˆ (ˆym) + (1 − φeˆ −φˆ − e−φˆ)v0(ˆym) = .1 (A.2) dxˆ Appendix A Mathematical Proofs 81

dyˆm 0 Using the equilibrium bidding condition dxˆ = u (ˆx), then

eφˆ − (eφˆ − φˆ − 1)v0(ˆym)u0(ˆx) φˆ0 = (A.3) φvˆ (ˆym)

Diﬀerentiating the buyer’s indiﬀerence condition (4.4)yields:

−φˆ0e−φˆu(ˆx) + e−φˆu0(ˆx) = 0 and u0(ˆx) φˆ0 = . (A.4) u(ˆx)

Then (A.3) and (A.4) together:

u0(ˆx) eφˆ − (eφˆ − φˆ − 1)v0(ˆym)u0(ˆx) = (A.5) u(ˆx) φvˆ (ˆym)

Under symmetric equilibriumx ˆ = x∗, φˆ = Φ :

Φv(ˆym∗) eΦ + (eΦ − Φ − 1)v0(ˆym∗) = , u(x∗) u0(x∗) withy ˆm∗ = u(x∗), and this completes the proof.

Proof of Proposition 4.5.

Proof. Totally diﬀerentiating the equilibrium condition,

Φv(u(x∗)) eΦ + (eΦ − Φ − 1)v0(u(x∗)) = . u(x∗) u0(x∗)

I have: v(u(x∗)) v(u(x∗)) dΦ + Φ ∂ ∂x∗ dx∗ + (eΦ − Φ − 1)v00(u(x∗))u0(x∗)dx∗ + (eΦ − 1)v0(u(x∗))dΦ = u(x∗) u(x∗) u00(x∗)eΦ eΦ − dx∗ + dΦ [u0(x∗)]2 u0(x∗) Appendix A Mathematical Proofs 82 then

v(u(x∗)) eΦ + (eΦ − 1)v0(u(x∗)) − dΦ u(x∗) u0(x∗) 00 ∗ Φ ∗ u (x )e Φ 00 ∗ 0 ∗ v(u(x )) ∗ ∗ = − + (e − Φ − 1)v (u(x ))u (x ) + Φ ∂ ∂x dx [u0(x∗)]2 u(x∗) or

n v(u(x∗)) 0 ∗ Φ h 0 ∗ 1 io ∗ dx u(x∗) − v (u(x )) + e v (u(x )) − u0(x∗) = − n 00 ∗ h ∗ io dΦ u (x ) + (eΦ − Φ − 1)v00(u(x∗))u0(x∗) + Φ ∂ v(u(x )) ∂x∗ [u0(x∗)]2 u(x∗) where

u”(x∗) < 0, v”(u(x∗))u0(x) < 0 u02(x∗) v(u(x∗)) v(u(x∗)) ∂ ∂x∗ = v0(u(x∗)) − u0(x∗) < 0; u(x∗) u(x∗)

The denominator is strictly positive.

Consider the numerator. Using the Largrange mean value theorem, I have:

v(u(x∗)) − v0(u(x∗)) > 0 u(x∗)

h 0 ∗ 1 i ∗ If v (u(x )) − u0(x∗) > 0 then x is an increasing function of Φ. Notice at pointx ¯1: v0(¯y1m)u0(¯x1) = 1

0 1 h 0 ∗ 1 i Since v (·) is decreasing, and u0(·) is increasing, v (u(x )) − u0(x∗) > 0 iﬀ x∗ < x¯1.

Combining terms together, I have:

dx∗ > 0 iﬀ x∗ < x¯1 dΦ Appendix A Mathematical Proofs 83

As x∗(Φ) is an increasing one-to-one reﬂection of Φ, and also I have already ∗ 1 ∗ mentioned, x (Φ) |Φ→∞→ x¯ . There is no way for x to reach the area of [¯x1, ∞).

I have: dx∗ > 0 dΦ

This completes the proof.

Cost-Production Structure The environment of our payoﬀ setting actually imbeds a very general class of cost-production structure. Presumably, for such a payoﬀ structure, the payment is not transferable but measurable, and the measure of such structure is presented by the cost-production structure. Given any cost-production structure that c0(x) > 0, c”(x) > 0, u0(x) > 0, u”(x) < 0, and u0(0) > c0(0), I denote c(x) asx, ˜ then dx˜ = c0(x)dx:

du u0(x)dx u0(x) = = > 0 dx˜ c0(x)dx c0(x) d2u c0(x)u”(x) − c”(x)u0(x) = < 0 dx˜2 (c0(x))3

0 u0(x) u0(x) as already known: c0(x) |x=0> 1, c0(x) < 0 and I defineu ˜ as the reflection x fromx ˜ to u(x), and definec ˜(˜x) =x ˜

u˜ :x ˜ → u then functionu ˜(˜x) satisfies:u ˜0 > 0,u ˜” < 0 andu ˜0 > 1,and since x → x˜ is a one-to-one, increasing reflection, for any givenx ˜, I can always map it back to x with x = c−1(˜x). n o The new payoff structure P˜ :x ˜ → (˜u(˜x), c˜(˜x)) is equivalents to the original structure {P : x → (u(x), c(x))}. Bibliography

[1] Daron Acemoglu and Robert Shimer. Eﬃcient unemployment insurance. Journal of Political Economy, 107(5):pp. 893–928, 1999. ISSN 00223808. URL http://www.jstor.org/stable/10.1086/250084.

[2] Nejat Anbarci and Ching-jen Sun. Asymmetric nash bargaining solutions: A simple nash programme. Economics Letters, 120(2):211–214, 2013.

[3] Kenneth J Arrow. The role of securities in the optimal allocation of risk-bearing. The Review of Economic Studies, pages 91–96, 1964.

[4] Kenneth Burdett, Shouyong Shi, and Randall Wright. Pricing and matching with frictions. Journal of Political Economy, 109(5):pp. 1060–1085, 2001. ISSN 00223808. URL http://www.jstor.org/stable/10.1086/ 322835.

[5] R. H. Coase. The nature of the ﬁrm. Economica, 4(16):386–405, 1937. ISSN 1468-0335. doi: 10.1111/j.1468-0335.1937.tb00002.x. URL http: //dx.doi.org/10.1111/j.1468-0335.1937.tb00002.x.

[6] Melvyn G. Coles and Jan Eeckhout. Indeterminacy and directed search. Journal of Economic Theory, 111(2):265 – 276, 2003. ISSN 0022-0531. doi: http://dx.doi.org/10.1016/S0022-0531(03)00086-3. URL http:// www.sciencedirect.com/science/article/pii/S0022053103000863.

[7] Gerard Debreu. Theory of value: An axiomatic analysis of economic equilibrium. Number 17. Yale University Press, 1959.

84 Bibliography 85

[8] Peter A Diamond. Wage determination and eﬃciency in search equilibrium. The Review of Economic Studies, 49(2):217–227, 1982.

[9] Peter A. Diamond. Aggregate demand management in search equilibrium. Journal of Political Economy, 90(5):pp. 881–894, 1982. ISSN 00223808. URL http://www.jstor.org/stable/1837124.

[10] Manolis Galenianos and Philipp Kircher. On the game-theoretic foundations of competitive search equilibrium*. International Economic Review, 53(1):1–21, 2012. ISSN 1468-2354. doi: 10.1111/j.1468-2354.2011.00669. x. URL http://dx.doi.org/10.1111/j.1468-2354.2011.00669.x.

[11] John C. Harsanyi and Reinhard Selten. A generalized nash solution for two-person bargaining games with incomplete information. Management Science, 18(5-part-2):80–106, 1972. doi: 10.1287/mnsc.18.5.80. URL http://dx.doi.org/10.1287/mnsc.18.5.80.

[12] Oliver Hart and John Moore. Incomplete contracts and renegotiation. Econometrica: Journal of the Econometric Society, pages 755–785, 1988.

[13] Benoit Julien, John Kennes, and Ian King. Bidding for labor. Review of Economic Dynamics, 3(4):619 – 649, 2000. ISSN 1094-2025. doi: http:// dx.doi.org/10.1006/redy.1999.0089. URL http://www.sciencedirect. com/science/article/pii/S1094202599900893.

[14] Jaehong Kim and Gabriele Camera. Uniqueness of equilibrium in directed search models. Journal of Economic Theory, 151(0):248 – 267, 2014. ISSN 0022-0531. doi: http://dx.doi.org/10.1016/j.jet.2013. 11.004. URL http://www.sciencedirect.com/science/article/pii/ S0022053113001944.

[15] Ricardo Lagos and Randall Wright. A uniﬁed framework for monetary theory and policy analysis. Journal of Political Economy, 113(3):pp. 463– 484, 2005. ISSN 00223808. URL http://www.jstor.org/stable/10. 1086/429804. Bibliography 86

[16] Benjamin Lester. Directed search with multi-vacancy ﬁrms. Jour- nal of Economic Theory, 145(6):2108 – 2132, 2010. ISSN 0022-0531. doi: http://dx.doi.org/10.1016/j.jet.2010.07.002. URL http://www. sciencedirect.com/science/article/pii/S0022053110000955.

[17] R Duncan Luce and Howard Raiﬀa. Games and decisions: Introduction and critical survey. Courier Dover Publications, 2012.

[18] Adrian Masters. Inﬂation and welfare in retail markets: Prior production and imperfectly directed search. Journal of Money, Credit and Banking, 45(5):821–844, 2013.

[19] R. Preston McAfee and John McMillan. Auctions and bidding. Journal of Economic Literature, 25(2):pp. 699–738, 1987. ISSN 00220515. URL http://www.jstor.org/stable/2726107.

[20] Espen R. Moen. Competitive search equilibrium. Journal of Political Economy, 105(2):pp. 385–411, 1997. ISSN 00223808. URL http://www. jstor.org/stable/10.1086/262077.

[21] Dale T. Mortensen. The matching process as a non- cooperative/bargaining game. Discussion Papers 384, Northwestern University, Center for Mathematical Studies in Economics and Manage- ment Science, 1979. URL http://EconPapers.repec.org/RePEc:nwu: cmsems:384.

[22] Dale T Mortensen and Christopher A Pissarides. Job creation and job destruction in the theory of unemployment. The review of economic studies, 61(3):397–415, 1994.

[23] John Nash. Two-person cooperative games. Econometrica: Journal of the Econometric Society, pages 128–140, 1953.

[24] John F Nash. The bargaining problem. Econometrica: Journal of the Econometric Society, pages 155–162, 1950. Bibliography 87

[25] Ed Nosal and Guillaume Rocheteau. Money, payments, and liquidity. MIT press, 2011.

[26] Martin J Osborne and Ariel Rubinstein. Bargaining and markets, vol- ume 34. Academic press San Diego, 1990.

[27] Michael Peters. Bertrand equilibrium with capacity constraints and restricted mobility. Econometrica, 52(5):pp. 1117–1127, 1984. ISSN 00129682. URL http://www.jstor.org/stable/1910990.

[28] Michael Peters. Ex ante price oﬀers in matching games non-steady states. Econometrica, 59(5):pp. 1425–1454, 1991. ISSN 00129682. URL http: //www.jstor.org/stable/2938374.

[29] Michael Peters. Equilibrium mechanisms in a decentralized market. Journal of Economic Theory, 64(2):390 – 423, 1994. ISSN 0022- 0531. doi: http://dx.doi.org/10.1006/jeth.1994.1074. URL http://www. sciencedirect.com/science/article/pii/S002205318471074X.

[30] Michael Peters. Limits of exact equilibria for capacity constrained sellers with costly search. Journal of Economic Theory, 95(2):139 – 168, 2000. ISSN 0022-0531. doi: http://dx.doi.org/10.1006/jeth.2000. 2667. URL http://www.sciencedirect.com/science/article/pii/ S002205310092667X.

[31] Michael Peters. The pre-marital investment game. Journal of Economic Theory, 137(1):186–213, 2007.

[32] Michael Peters and Aloysius Siow. Competing premarital investments. Journal of Political Economy, 110(3):592–608, 2002.

[33] C. A. Pissarides. Eﬃcient job rejection. The Economic Journal, 94:pp. 97–108, 1984. ISSN 00130133. URL http://www.jstor.org/stable/ 2232658.

[34] Christopher A Pissarides. Equilibrium unemployment theory. MIT press, 1990. Bibliography 88

[35] Richard Rogerson, Robert Shimer, and Randall Wright. Search-theoretic models of the labor market-a survey. Working Paper 10655, National Bureau of Economic Research, July 2004. URL http://www.nber.org/ papers/w10655.

[36] William P. Rogerson. Contractual solutions to the hold-up problem. The Review of Economic Studies, 59(4):777–793, 1992. doi: 10.2307/ 2297997. URL http://restud.oxfordjournals.org/content/59/4/ 777.abstract.

[37] Ariel Rubinstein. Perfect equilibrium in a bargaining model. Economet- rica, 50(1):pp. 97–109, 1982. ISSN 00129682. URL http://www.jstor. org/stable/1912531.

[38] Ariel Rubinstein, Zvi Safra, and William Thomson. On the interpretation of the nash bargaining solution and its extension to non-expected utility preferences. Econometrica: Journal of the Econometric Society, pages 1171–1186, 1992.

[39] Thomas C Schelling. The strategy of conﬂict. Harvard university press, 1980.

[40] Cemil Selcuk. Trading mechanism selection with directed search when buyers are risk averse. Economics Letters, 115(2):207–210, 2012.

[41] L.S. Shapley and M. Shubik. The assignment game i: The core. Interna- tional Journal of Game Theory, 1(1):111–130, 1971. ISSN 0020-7276. doi: 10.1007/BF01753437. URL http://dx.doi.org/10.1007/BF01753437.

[42] Alberto Trejos and Randall Wright. Search, bargaining, money, and prices. Journal of Political Economy, 103(1):pp. 118–141, 1995. ISSN 00223808. URL http://www.jstor.org/stable/2138721.

[43] Makoto Watanabe. Inﬂation, price competition, and consumer search technology. Journal of Economic dynamics and control, 32(12):3780– 3806, 2008. Bibliography 89

[44] Makoto Watanabe. A model of merchants. Journal of Economic Theory, 145(5):1865–1889, 2010.

[45] Makoto Watanabe. Middlemen: A directed search equilibrium approach. Technical report, Tinbergen Institute Discussion Paper, 2012.